ssh supports X11 "tunneling". This is useful because if the Pi (or any remote computer) requires traversing the Internet, you wont need to worry about opening up firewall ports or configuring NAT rules.
If you are not traversing a network where tunneling is needed, then you don't need to worry about ssh (more on that later). But since your question was about PuTTY & ssh I'll address that first.
If using PuTTY, you enable this by navigating to category
Connection -> SSH -> X11 and make sure you tick the box labeled "Enable X11 forwarding".
You also need to tell PuTTY where to find your local X11 server (this is the address of the Xming server). The default for this will be
X11 is a network-enabled display protocol.
The nomenclature here is
hostname can be any valid host or IP address.
Display is the X11 Display number (an integer) and usually this is '0'.
Screen is the screen number (an integer) on that display and also is usually '0' and can be omitted if the X11 Display only has one screen.
(For those using a Mac or Linux box and not using Putty, you can do X11 tunneling by including the '-Y' flag in your ssh command e.g. ssh -Y user@host).
This requests that the remote computer start an X11 forwarder. It will create a virtual X11 display that appears to run on localhost:10.0 (it will use "10" if it is the first ssh session with X11 forwarding. If another ssh client connects and also requests X11 forwarding then it will get localhost:11, and so on.
After you log into your Pi via PuTTY (or ssh command-line) verify that the DISPLAY environment is set:
You should see output like this:
Now run anything with a GUI ... such as galculator and verify that it displays.
All of the above assumed you actually needed ssh tunneling of X11 (usually because you're traversing the Internet). If the Pi and your PC are on the same network (ports are not blocked) then you don't actually need to do anything special with ssh... just connect as you normally do.
BUT... you need to manually set your DISPLAY environment variable. This will be set to the hostname of your PC and the X11 server number.
Note that I used the X11 server & display numbers of :0.0 (and could have just used :0). This assumes you only have one Xming server running and it picks its default server number.
You may have a few more issues to deal with.
First, X11 has it's own security. It normally will only accept a local connection. To get it to accept remote connections you could either display disable access control completely (dangerous!) or you can add the hostnames (or IP addresses) to your X0.hosts file. This is usually C:\Program Files (x86)\Xming\X0.hosts (and you must be Administrator to edit it). Creating a specific list of permitted hosts is much safer than just switching off access control completely.
You may need to adjust Windows firewall rules to allow Xming to run and accept connections when on a private network (allowing this over a public network isn't advisable ... that's where ssh tunneling is a much safer option.)