Attending conferences is a vital part of being a scientist. By listening to talks, you find out who the real stars in the field are and get the latest results that you may not have been aware of. By giving talks (and to a lesser extent posters), you show the world who you are, other than yet another name on astro-ph. Finally, after hours you can network with your peers and with potential future job prospects alike.
There are both disciplinary conferences and general conferences. The reasons for attending the two differ. Disciplinary conferences are smaller events but tend to draw most people working in your specialty of choice, so once you're entrenched in a subject it becomes extremely important to attend these to secure your reputation with the people that matter as well as keep up-to-date with all the newest results that might affect your research immediately. In contrast, general conferences are most useful for getting a broader perspective of what's happening outside your sub-field and understanding the broader context of how what you are doing is relevant to astronomy in general, though there are usually discipline-specific tracks there as well. These tend to be very large meetings.
Never been to a conference before? Nervous about speaking? Presenting a poster at the American Astronomical Society conference is a great way to share your work in a student-friendly setting! http://aas.org Conferences happen in the winter and summer.
A list of many (all?) astronomy related conferences is maintained at http://www1.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/meetings/
A list of many (all?) EXOPLANET conferences is here: http://exoplanet.eu/meetings/
The major general conference in the United States is the semi-annual meeting of the American Astronomical Society. The winter meeting tends to be much larger than the summer meeting and is the most useful to attend. For more information, click below.