Most people outside of astronomy have little idea how telescope time is assigned and used. Contrary to popular fiction, astronomers generally do not live on mountain tops, sleeping all day and working all night by staring through the eyepiece of a telescope. Except for the few astronomers hired to run telescopes, most of us work fairly normal hours at universities. We teach classes and do our research on computers, only rarely travelling to the mountains to use telescopes.
So, how do we get telescope time? The first part involves writing a proposal. We have to answer several questions. What science do we want to do? What star or galaxy do we want to look at? Why do we want to use this particular telescope? What cameras will we be using? How much time do we need to finish the science? What are the acceptable dates and times to use the telescope? And can we back all of this up with calculations?
Once each astronomer has written her/his proposal(s), the proposals are collected and sent to a "Time Allocation Committee," or TAC. The TAC must then read through the proposals and rank them. The rankings include a mix of how important the science is and how likely the project is to succeed. There are always proposals with great scientific ideas, but the proposed observations seem quite unlikely to succeed. Sometimes a project will definitely work, but the science isn't that exciting.
The TAC's job is a thankless task, because usually there are two to ten times more time requested than actually exists. So many good projects that are likely to succeed won't get time just because there isn't enough to go around. And how do you decide who gets time and who doesn't? It's pretty arbitrary a lot of the time.
Once the TAC is done, there is one last step -- the actual scheduling. The people who schedule the telescope use the TAC's rankings to guide their work. Often they can't schedule all of the highest-ranked proposals, though. Suppose there are two highly-ranked projects that will only succeed on May 23. Then the scheduler has to make a tough choice!
Anyway, that's how we get our time. It's a hard process, and even the best astronomers won't get all the time they want. But it pushes us to think hard about our projects and what we need to succeed.