Check what storage an app uses? Is there a way to check what storage a third party application is using? i.e  which of these it uses?
If you are rooted, this is quite easy to find out.

First you need to find the app's package name, e.g. com.swype.android.inputmethod for Swype, or com.google.android.apps.maps for Google Maps, for example.  You can usually search for the app in Android Market, and the link will contain its package name, e.g. https://market.android.com/details?id=com.google.android.apps.maps would be the link to Google Maps.  Usually the package names are descriptive enough that you can simply guess which app they represent just by looking at it.

You can do a few simple ls commands in Terminal Emulator or ADB shell to find out the app's name, and where it's stored.  However, you will need to have root access to the device since app and data storage is protected.

Pre-installed apps are generally located in /system/app directory.  Use the command ls -l /system/app/ to find how much space a particular system app is taking up:

ls -l /system/app/Browser.*
-rw-r--r-- root     root               418098 2011-06-26 01:05 Browser.apk
-rw-r--r-- root     root               472760 2011-06-26 01:05 Browser.odex


On most Stock ROMs you will see an .odex file along with a .apk file for each app - those are pre-compiled binaries optimized for your given device.  You cannot remove them.

3rd-party apps are installed to /data/app or /data/app-private directories.  Command ls -l /data/app/[package_name_here]* should return the amount of space taken up by this app on internal memory.  For example, here's the output of looking at Google Maps apk:

# ls -l /data/app/com.google.android.apps.maps*
ls -l /data/app/com.google.android.apps.maps*
-rw-r--r-- system   system            6330104 2012-02-04 09:01 com.google.android.apps.maps-2.apk


The app data and settings are located on /data/data/ partition.  There you will find a directory for each installed app, e.g. /data/data/com.google.android.apps.maps/ for preferences and settings related to the Google Maps app.  Again, ls -l /data/data/com.google.android.apps.maps/* will return a listing of all files with their sizes:

# ls -l /data/data/com.google.android.apps.maps/*
ls -l /data/data/com.google.android.apps.maps/*
-rw------- app_189  app_189              3783 2012-02-07 11:29 mobilemaps.clients.google.com.443
-rw------- app_189  app_189                37 2012-02-01 16:14 nlp_GlsPlatformKey
-rw------- app_189  app_189               208 2012-02-07 11:21 macs
drwxrwx--x app_189  app_189                     2012-02-01 16:14 webviewCache
-rw------- app_189  app_189              2450 2012-02-07 11:21 nlp_state
drwx------ app_189  app_189                     2012-02-01 16:14 selectors
-rw------- app_189  app_189               290 2012-02-07 11:21 nlp_devices
drwx------ app_189  app_189                     2012-02-01 16:14 models
-rw-rw---- app_189  app_189             18432 2012-02-03 13:29 google_analytics.db
-rw-rw---- app_189  app_189              5120 2012-02-07 10:38 google_latitude.db
-rw-rw---- app_189  app_189              6144 2012-02-01 16:14 webviewCache.db
-rw-rw---- app_189  app_189              5120 2012-02-05 17:27 local_active_places.db
-rw-rw---- app_189  app_189              4096 2012-02-05 17:27 da_destination_history
-rw-rw---- app_189  app_189              5120 2012-02-05 17:49 LayerInfo
-rw-rw---- app_189  app_189              5120 2012-02-05 17:26 search_history.db
-rw-rw---- app_189  app_189              6144 2012-02-03 13:29 userfeedback.db
-rw-rw---- app_189  app_189             14336 2012-02-01 16:14 webview.db
-rw-rw---- app_189  app_189                38 2012-02-07 11:13 DATA_SYNC_DATA_LOCAL
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_145479516184369608
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_4406538396446325405
-rw-rw---- app_189  app_189              1813 2012-02-05 13:48 DATA_PROTO_CLIENT_SAVED_PHOTO_CACHE
-rw-rw---- app_189  app_189               338 2012-02-06 23:41 DATA_TILE_HISTORY
-rw-rw---- app_189  app_189                12 2012-02-05 17:49 DATA_SAVED_REMOTE_ICONS_DATA_BLOCK
-rw-rw---- app_189  app_189             36580 2012-02-01 16:14 DATA_RemoteStringsBlock_en
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_240647554975296207
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_443613581827136957
-rw-rw---- app_189  app_189              5620 2012-02-05 17:49 DATA_SAVED_BGFS_3
-rw-rw---- app_189  app_189                57 2012-02-05 17:49 DATA_STARRING
-rw-rw---- app_189  app_189               573 2012-02-05 10:04 NavZoomTables.data
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_-7991748045000310303
-rw-rw---- app_189  app_189               217 2012-02-05 15:29 DATA_LAYER_4
-rw-rw---- app_189  app_189             26756 2012-02-01 16:14 DATA_PHOTO_-4352347968798340772
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_-1960244636679913737
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_7546641620697631512
-rw------- app_189  app_189                34 2012-02-07 11:21 nlp_clts
-rw-rw---- app_189  app_189              2492 2012-02-07 11:30 DATA_location_history
-rw-rw---- app_189  app_189                11 2012-02-01 16:14 DATA_LAYER_0
-rw-rw---- app_189  app_189              1186 2012-02-01 16:14 ZoomTables.data
-rw-rw---- app_189  app_189                11 2012-02-05 17:49 DATA_SAVED_BGFS_EXTRA_3
-rw-rw---- app_189  app_189             18994 2012-02-05 17:49 DATA_PROTO_SAVED_CATEGORY_TREE_DB
-rw-rw---- app_189  app_189               247 2012-02-05 17:49 event_store_v2_driveabout
-rw-rw---- app_189  app_189              5294 2012-02-05 10:05 DA_LayerInfo
-rw-rw---- app_189  app_189               380 2012-02-05 16:23 DATA_LAYER_5
-rw-rw---- app_189  app_189             11387 2012-02-05 17:49 DATA_SYNC_DATA
-rw------- app_189  app_189                35 2012-02-07 11:21 cp_state
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_-6319433669272599441
-rw-rw---- app_189  app_189                76 2012-02-01 16:14 DATA_ServerControlledParametersManager.data
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_-4787236618283354619
-rw-rw---- app_189  app_189                57 2012-02-01 16:14 DATA_star_details_index
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_-7325828578927889496
-rw-rw---- app_189  app_189                76 2012-02-05 17:27 DATA_ServerControlledParametersManager_DA.data
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_-5041077772587057686
-rw-rw---- app_189  app_189                42 2012-02-05 17:26 DATA_PROTO_SAVED_RECENT_LAYERS
-rw-rw---- app_189  app_189                21 2012-02-05 10:04 DATA_SAVED_BGSF_
-rw-rw---- app_189  app_189               263 2012-02-05 11:30 DATA_LAYER_2
-rw-rw---- app_189  app_189                26 2012-02-01 16:14 NavigationParameters.data
-rw-rw---- app_189  app_189               330 2012-02-05 17:27 DA_DirOpt_en_US
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_-4817178430903221341
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_8476874153932292760
-rw-rw---- app_189  app_189            141298 2012-02-05 17:27 DATA_Restrictions
-rw-rw---- app_189  app_189                29 2012-02-05 17:49 DATA_PROTO_SAVED_LAYER_STATE
-rw------- app_189  app_189                61 2012-02-01 16:14 nlp_params
-rw-rw---- app_189  app_189                 0 2012-02-05 17:27 DATA_Restrictions_lock
-rw-rw---- app_189  app_189             26756 2012-02-01 16:31 DATA_PHOTO_8041021750103678571
-rw-rw---- app_189  app_189             10237 2012-02-03 13:28 DATA_star_details_0
-rw-rw---- app_189  app_189                 6 2012-02-05 17:49 DATA_RECENT
-rw-rw---- app_189  app_189              5048 2012-02-06 23:41 DATA_Preferences
-rw-rw---- app_189  app_189                65 2012-02-01 16:14 network_initiated_prefs.xml
-rw-rw---- app_189  app_189                65 2012-02-01 16:14 friends_app_widget_preferences.xml
-rw-rw---- app_189  app_189              1269 2012-02-01 16:14 ids.xml
-rw-rw---- app_189  app_189               250 2012-02-03 13:27 settings_preference.xml
-rw-rw---- app_189  app_189                65 2012-02-01 16:14 cache_settings_preference.xml
-rw-rw---- app_189  app_189               358 2012-02-07 11:28 checkin_notification_preferences.xml
-rw-rw---- app_189  app_189               119 2012-02-01 16:14 com.google.android.apps.maps_preferences.xml
-rw-rw---- app_189  app_189               345 2012-02-05 17:49 DriveAbout.xml
-rw-rw---- app_189  app_189               317 2012-02-07 10:38 LOCATION_REPORTING.xml


Android 2.2 (Froyo) introduced the ability to move apps to SD card (or "external storage" on some devices).  This introduces a level of complexity since portions of the app get moved around.  For every app that you move to SD Card/media area, the system creates a virtual mount point named /dev/block/dm-X, where "X" is a sequential number.  It them moves the app's APK (Android package) file and some preferences there.  You can view this by issuing a mount command in shell/terminal emulator.  Along with other usual mounts, it should show which apps were moved to SD Card like in an example below:

/dev/block/dm-0 /mnt/asec/com.metago.astro-1 vfat ro,dirsync,nosuid,nodev,relatime,uid=1000,fmask=0222,dmask=0222,codepage=cp437,iocharset=iso8859-1,shortname=mixed,utf8,errors=remount-ro 0 0
/dev/block/dm-1 /mnt/asec/com.a0soft.gphone.aTrackDog-1 vfat ro,dirsync,nosuid,nodev,relatime,uid=1000,fmask=0222,dmask=0222,codepage=cp437,iocharset=iso8859-1,shortname=mixed,utf8,errors=remount-ro 0 0


Here I moved Astro file manager and aTrackDog apps to the SD Card on my phone.  Use ls /mnt/asec/ to view the directories created for the moved apps.  Issuing command ls -l /mnt/asec/* will list the file sizes within those folders:

# ls -l -a /mnt/asec/*
ls -l -a /mnt/asec/*
-r-xr-xr-x system   root               332826 2012-02-07 11:21 pkg.apk
dr-xr-xr-x system   root                        2012-02-07 11:21 lib
-r-xr-xr-x system   root              2363522 2012-02-07 11:18 pkg.apk
dr-xr-xr-x system   root                        2012-02-07 11:18 lib


The wrapper for the virtual partition gets stored inside a hidden ".android_secure" folder of the SD Card:

# ls -a -l /mnt/sdcard/.android_secure
ls -a -l /mnt/sdcard/.android_secure
----rwxr-x system   sdcard_rw          4290560 2012-02-07 11:18 com.metago.astro-1.asec
----rwxr-x system   sdcard_rw          2161664 2012-02-07 11:21 com.a0soft.gphone.aTrackDog-1.asec


Once moved the app's APK will no longer be under /data/app/ or /data/app-private/ folders.

Another area of interest is Dalvik Cache - an area that contains cached data for each installed app.  Upon first boot-up or new app installation the system generates a .dex file for each app that it uses for quicker start-up.  These files are stored on /data/dalvik-cache/ partition.  Command ls -l /data/dalvik-cache/*[package_name_here]* will return the space taken up by the app's Dalvik cache.  For example, here's how much space is taken up by Google Maps app:

# ls -l /data/dalvik-cache/*com.google.android.apps.maps*
ls -l /data/dalvik-cache/*com.google.android.apps.maps*
-rw-r--r-- system   app_189           4406016 2012-02-04 09:01 data@app@com.google.android.apps.maps-2.apk@classes.dex


Next, the /cache partition of the internal memory may be used by apps.  The command ls -l /cache/ will show you all files stored there, along with sizes and owner information.  For example, on my phone here's the output:

# ls -l /cache/
ls -l /cache/
-rw-r--r-- app_13   app_13            3640954 2012-02-07 09:16 downloadfile-1.apk
drwxrwx--- system   cache                       2011-12-31 21:03 recovery
drwxrwx--- root     root                        2011-12-31 21:00 lost+found


Besides the standard directories, there is a file named downloadfile-1.apk that belongs to something named "app_13".  I can cross-reference this "app_13" name to the listing of ls -l /data/data/ to see that this file belongs to Android's built-in downloader:

# ls -l /data/data/
ls -l /data/data/
......
drwxr-x--x app_13   app_13                      2011-12-31 21:02 com.android.providers.downloads.ui
......


Finally, apps can use the SD Card for their own purposes, and there is really no way to track this, unless the app names its folders something recognizable.