It sounds like you answer primarily to programmers, so my answers may not apply as directly as they would to a regular Microsoft Office-using Accountant, Marketing or HR rep.
Solve problems - 99% of people want their computer problems solved so they can continue doing their job. Solving the problem means actually fixing the specific problem they asked about, giving them a workaround or explaining that you don't know how to or don't have the resources to fix the problem. Everything else flows from this. When you act like you don't care, or forget about people's problems, or don't respond at all, they get upset.
Explain why - If you're going to say "No," explain why. If you're too busy to help, explain why. If the user's request is impossible or infeasible, explain why. No one likes a dictator.
Speak to people - IT folks like to have conversations via e-mail. Talking to the person (preferably face-to-face) gets things done a lot more efficiently and makes people feel that you care about their problem.
Listen and explain - don't cut people off or assume you know what the problem is. Listen to what they have to say, and explain why the problem occurred and what to do about it in the future.
Respond quickly - People get justifiably upset when you don't respond to their requests for weeks (or more). I'm not a fan of SLAs, but even if the response is "I don't have time right now," it's better than no response at all.
Be flexible - Don't let No be the first thing out of your mouth every time someone asks for something. Your job is not to control technology. It's to better enable people to do their jobs. Work with your users, not against them.
Follow up - Don't let requests slip through the cracks. People get upset when they can't do their jobs and feel like they're being ignored. Complete or respond to every request, every time, even if the answer is "That's not possible."
Be friendly - People's cubicles say a lot about them. If they have bicycling photos, ask them about bicycling. If they're stamp collectors, ask about stamps. End user support is 80% customer service and 20% technical (though this ratio is different for programmers as end-users).
Don't be condescending - The classic IT mistake is to confuse an inability to use a computer with stupidity. When you talk down to someone because you think they're stupid, they know. When you go back and degrade someone to your coworkers, they can tell. You don't know anything about the 8,342 forms in HR (or at least I don't), but that doesn't make you an idiot. So don't treat your end-users that way. Also, don't act like they're bothering you with their stupid requests. This is probably how your cable company treats you - and how much do you like your cable company?
Communicate maintenance - If you're making changes to anything that has a chance of creating a problem, tell people. It's impossible to know every deadline in every department. If your maintenance makes someone miss their deadline, they're going to be upset. When you make changes that no one knows about and they break things, you're giving people the impression that you don't care.
Solicit feedback - Do you know what your users pain points are? Do you know how you can improve service? Ask them. It may get management to hire that extra employee you need.
Encourage followups - Encourage the user to call you back if the problem wasn't fixed the first time.