I've rooted my phone.  Now what?  What do I gain from rooting? This is a common question by those who have just rooted their phones.  What apps, ROMs, benefits, etc. do I get from rooting?  What should I be doing now?
Things that Require Root
Root File Explorers
You can remount your /system/ directory read-write, which will allow you to remove carrier-shipped applications you don't like and so on.  Root file explorers can allow you to view all folders on your phone, such as /data/ which is normally protected.

Mount /system RO/RW
Root Explorer
ES File Explorer

Patching Apps
With root access you can patch other apps to gain special access to functions, or removal of self-protection for some apps.

Freedom - Get iAP (in-App Purchase) for free! Buy diamonds, rubies and Pro unlocks without real money. Note: Legal restrictions may apply depending on your country/region.
Lucky Patcher - Patch apps so that they don't require Google Play Services / displays no more ads / automatically become Pro version / work at certain DPI... There's many you can do with Lucky Patcher and there's even a bunch of patches provided already.

Moving &amp; Removing Apps
Access to /system also means you can move apps or updates to system apps from /data to /system (with caution!), if the partition has enough space.

Titanium Backup - Back up all of your applications and data.  Also allows you to uninstall or freeze system apps, integrate updates into your ROM to save space, move apps to SD, and more.
apps2rom - does for free what the pro version of titanium backup does: lets you move apps to the system partition.
Bloatfreezer - Freeze or remove system apps
Link2SD - Move (system) applications to a second partition in SD card, liberating loads of free space in /system partition.

Note that moving apps to the sdcard has risks and cons, and should be undertaken only if space is needed. Most modern devices have enough memory for many apps (16GB+), providing that photos/videos/music are stored on an external sdcard.
Tethering
Some devices already have Wi-Fi tethering out-of-the-box, like the Samsung Galaxy S, so this isn't such a big deal on them unless the carrier charges for the feature.

Android Wi-Fi Tether
Wireless Tether

Screenshots &amp; screen recording
root is needed for screenshots before 4.0-Ice Cream Sandwich.
Most pre-lollipop screen recorders also require root.

ShootMe
Drocap2

Backups

Titanium Backup - Back up all of your applications and data.  Also allows you to uninstall system apps, integrate updates into your ROM to save space, and so on.
ClockworkMod Recovery (ROM Manager) - Usually requires root to install via ROM Manager.  Allows you to backup the entire device, including your ROM, as well as flash new ROMs and other advanced tasks.

There are also many apps that claim to make a full nandroid backup.
Application/Network Control

AdFree - Removes ads by changing the hosts file (so both ads in apps and in browser)

Adaway - Also blocks in-app adds.

Droid Wall - Firewall application.  Got an app that's using up all your data when you're not even using it?  Block it with this!

Orbot - Tor app

Cache Mate - Clear cache of all apps

Droid VNC Server - Connect to your phone via VNC

LBE Security Master - Selectively revoke permissions from app

Button Savior - Helps to add on screen button to compensate a failure of a physical button.

ClockSync - to automatically sync the device's time with few predefined atomic clocks.


Xposed Framework
The Xposed framework allows apps with advanced tweaking functionality to be installed.

GravityBox (for Android 4.2 4.4 5.0 5.1 6.0) - An app that provides many system tweaks.
XInstaller - Various tweaks for Package Installer.
XHangouts - Improved version of google hangouts.
XWhatsApp - Mod for the popular whatsapp messenger.
XPrivacy - Finegrained app control.

CPU Control

SetCPU - Control the processor speed of the phone, can also setup profiles.  It allows underclocking based on conditions, e.g. underclock to 125mhz when screen is turned off. It dramatically increases battery life without affecting performance (when you're actually using apps / the phone) or can improve performance via overclocking.
Overclock Widget - Similar to SetCPU but less functional

Enable swap file / partition

ROEHSOFT RAM-EXPANDER (SWAP) - Use SD card for more RAM.

Memory Swapper Free - support swap file.


Advanced and Miscellaneous

ROM Manager - Facilitates installing new ROMs with ClockworkMod Recovery, and allows you to back up your entire device (ROM + apps + data) and other advanced tasks.
Quick Boot - Allows you to reboot into recovery or the bootloader / Download Mode.
LED Hack - Disable LED on phone (no longer available on Android Market)
Lag fixes - Various tweaks and filesystem changes to speed up your phone or prevent issues with your phone pausing / freezing temporarily.  These are particularly useful on some Samsung phones that use an RFS filesystem.  An Ext4 lagfix like Voodoo is recommended, but there are also Ext2 lagfixes like RyanZA's One Click Lag Fix which also allows you to change advanced memory, Wi-Fi, and disk scheduling options.
TRIM - Some older kernels do not have TRIM.
StickMount - Automatically mount USB flash drives, keyboards, mouse via USB OTG cable.
BusyBox - Makes terminal emulators feel more like linux.
Sideloading apps.  Most carriers/phones allow this option but AT&amp;T does not.

ROMs
Though root is not required for flashing new ROMs, many apps that make it easier do require root.
CyanogenMod is a very popular rom that many people put on their phones.  It is also easy to install via the ROM Manager app.  First install the Clockwork recovery.  Then run a backup before you flash any roms.
Also check out &quot;Where can I find stock or custom ROMs for my device&quot; or the XDA forums for other custom roms.  Most devices have a specific &quot;Android Development&quot; sub-forum where ROMs are posted.
Always do a nandroid backup before installing any rom or mod!  You may also need to wipe all data and cache from your phone before installing or upgrading a ROM.
Custom Kernels
One popular set of kernels is those provided by ChevyNo1. You can also download them via the premium version of the ROM Manager.  Make a nandroid (ClockworkMod) backup before using these kernels! You'll also want to get SetCPU to make the most out of these kernels.
Start with the low voltage kernels at the lowest speed and work your way up to the 1.2GHz.  If your phone is stable up to the 1.2GHz range, then try some of the ultra low voltage kernels.  If you start getting force closes, then switch back to a low voltage kernel.
Basically each phone (of the same phone brand/model) varies by which kernel it can handle due to the manufacturing differences between processors. So I may have a Motorola Droid that can run ultra low voltage kernels and yours may not be able to run them.  These phones weren't necessarily designed to run like this.