Browse for Servers does not display local instance
Your title says you have an installation problem, but I don't think this is the case (it seems like you have installed SQL Server okay, you just can't find it). Of course I don't know exactly what options you selected during setup, but from the programs applet, it seems like you should have at least one instance installed.

The Browse for Servers dialog is hokey and won't always find all instances, even on your own computer, even on the best days. I would rather use Configuration Manager to determine what's installed. Find the most recent version of this, since older versions of the Configuration Manager won't be able to see/manage newer versions of SQL Server (but the older versions of Config Manager don't get replaced or uninstalled).



If you can't find Configuration Manager, you can add it to any generic MMC console. Run mmc from the command line:

mmc.exe


(If you have that awful thing called UAC enabled, you may have to give the tool permission to run.)

Then choose file > add/remove snap-in:


  


Then pick SQL Server Configuration Manager and click Add (if there are multiple, they are from different versions, but not labeled correctly, so add all of them), then click OK.


  


Then open SQL Server Services, and you will see the list of instances on the right. If you had to add multiple nodes, make sure you check all of the nodes, since it is impossible to tell visually which node represents which version of Config Manager.


  


If that list is empty, you did not install SQL Server correctly.

To connect to one of those, first you need to make sure it's running; if not, right-click and select Start:


   


Now in Management Studio you should be able to connect by entering the right instance name. It will be in this form:

(local)\instance_name_from_within_brackets_in_the_list


So, once I have started that instance picture above, I would enter the following in the SSMS connect to server dialog:

(local)\SQL2016_407


Unless the instance name in brackets is MSSQLSERVER - this is a special display name reserved for the default instance. In this case you should be able to connect using just (local).

There are some other exceptions as well - for example if you only installed LocalDB, this is not going to show up as an active instance in Configuration Manager - this is by design (the whole point of LocalDB is to remove instance-level management). For more information on using LocalDB, see this tip.