Rocket Surgery Made Easy by Steve Krug is the golden standard to read here, especially if you don't have a lot of experience and need to get something done fast.
The process tends to be:
Look at your product and see what needs to be tested, prepare some tasks and situations to test.
Find participants that fit your target audience -- don't underestimate this, it can be harder than it seems.
Prepare a pre-test questionnaire for getting to know the user, his goals, his needs, his environment, and his experience related to your product.
Prepare a post-test questionnaire for things with questions you'd like to ask after the tasks are done (e.g. about the product in general). Keep in mind that there'll probably be additional questions that come up during your testing -- e.g. asking to clarify some behavior from before.
If you don't have much experience, I'd recommend testing with a colleague (or friend) or two before you get to the actual test.
During the test, it's very important to encourage the user to think out loud and to not give any hints about behavior. This can be harder than it seems, especially when you need to ask for clarification about behavior without giving away what the expected behavior was. (Sometimes, these questions are best kept for the end of the test.)
It's best to conduct the test with someone else being the main note taker, as you'll be busy interacting with the participant. Take notes too, though. Be sure to record the test (a screen + microphone recording will suffice, though sometimes it's useful to see the participant's facial expressions too.) and timestamp the notes you take.
You might be surprised that problems pop up in places where you didn't expect them to -- be ready for that and make sure to ask for clarification, either right away or at the end of the test (depends on the situation).
That's the jist of it really. It's not hard, but you do have to do it a few times to really get the hang of it.
ADDENDUM: I left out a few important points:
You shouldn't be the only one to observe the test -- at least one more observer should be present. (Krug recommends streaming the recording of the test live in another room to the team behind the tested application, but that's not always doable.)
It's good to briefly analyze your test right after it's done while it's still fresh in your mind, comparing notes with the other observer(s) and looking for emerging patterns in the tests done so far.
Additionally, I only mentioned traditional live user tests, but it's also possible to do testing online. You can do live streaming (using videoconferencing tools), but that can be a hassle to set up. There are also services where you provide the tasks and questions, which are then relayed to some people from the target demographic. I have experience with one non-real-time service -- the disadvantages are that you can't pose additional questions that come up in the process and you can't see the participants' facial expressions. It is easier to organize, though.