The Census Bureau provides detailed tables for the 2012 and 2010 elections, including information on reported voting and registration for various demographic and socioeconomic characteristics. This is part of the Current Population Survey, a monthly survey of approximately 50,000 households. Note that this survey does not include questions regarding party identification or candidates supported.
You can also access historic tables on reported voting and registration for various demographic and socioeconomic characteristics, with data back to the 1960s-1970s depending on the question.
For elections from 2006-present, use ANES to research voter characteristics in presidential year elections (not midterms).
Exit polls are conducted after voters have cast their ballots, and involve asking voters which candidates they voted for. You can use exit polls to understand the composition of voters in a given election and to make comparisons across years. In the US, the private firm Edison Research conducts exit polls for the major television networks, in what is known as the National Election Pool Consortium. You can view their exit poll results via CNN (linked below) and other network websites.
What you can learn from exit polls:
Exit polls can give you a picture of the electorate in terms of gender, age, race, party identification, ideology, education, income, religion, and other demographic factors. These polls also include questions that pertain to voters' opinions on political issues such as health care reform, questions that ask what voters look for in a candidate, and questions about voters' opinions about particular candidates and officials decisions.
It's important to acknowledge the limitations of exit polls and to critically evaluate your conclusions in light of these limitations. Exit polls are surveys, and although interviewers are instructed to survey a random sample of voters, the process is far from perfect due to a variety of factors.
If you plan to use exit poll data, reading Mark Blumenthal's Exit Polls: What You Should Know is a good place to start. (If you're truly pressed for time, Nate Silver has a shorter post that largely restates Blumenthal's main points.)