Once you have picked up the Aperture Science Handheld Portal Device (or so-called portal gun), you will need to find a portal-able surface (read portal-able surface section below) and left-click using your mouse to create a blue portal.
You will only be able to fire a blue portal with the Aperture Science Handheld Portal Device if it is not upgraded.

After a bit of intense gaming you should have now acquired a fully-upgraded Aperture Science Handheld Portal Device. This means that you can fire two different portals.
To fire your second portal, hold your mouse and right-click at a portal-able surface. This should cause an orange portal to appear on the surface, and it may connect to a blue portal if already placed.
You can also refer back to the section above if you are struggling to fire your blue portal.

White surface = Portal-able
Black surface = Not portal-able

Sometimes surfaces outside of test chambers such as behind-the-scenes areas may have office walls that are portal-able (they are also white walls).

Congratulations!

You are now a professional at using the Aperture Science Handheld Portal Device!

Use your power wisely and remember to keep your mouse connected/charged