If you're interested in researching birth records, there has never been a better time in history to do so. There is a wealth of information available on the Web now, including archived information, primary sources, and pointers to offline records.
The most reliable source for birth records are primary sources; i.e., the originating entities that actually processed the documents. Birth certificates and records are materials that are authenticated by governmental and hospital organizations. Obtaining copies of birth records varies by state; if you're attempting to get a recent birth certificate (say in the last fifty years), your best bet is to contact the originating entity and go from there. In this case, unfortunately, the Web is not going to be of much help, other than pointing you to your state's vital records office!
If you're looking for material that is not necessarily recent, than the Web is going to be extremely helpful. Some data is not available online simply because it hasn't made its way to the Web quite yet; for instance, census records aren't available to the public for at least a few decades after their first release.
One of the best sources online for birth certificates and other vital records is FamilySearch, a genealogy service maintained by the Church of Jesus Christ of Latter-Day Saints. You do not have to be a member of the church to access the site. The search function includes everything someone researching their genealogy will want to find: birth records, death records, census data, marriage, etc.
You'll need to have a first and last name, at least, in order to get your search going. The more information you know the better your search will be; for instance, enter in the country and state, if you know what it is, and that will definitely serve to narrow down your results. I wouldn't recommend checking off the "Match All Terms Exactly" box; that makes your search too restrictive (at least at first).
Your search results will come back with US Census information, user-submitted genealogies, and a multitude of search filters on the lefthand side that you can use to further narrow your results. Different filters will give you different levels of information, and it's smart to toggle these to come up with different combinations of information. Original records are available here to view, and it's absolutely fascinating to page through records that are hundreds of years old right within your Web browser.
What if I want to find more recent birth records?
Birth records are kept safe in the archives of state offices. The easiest way to track down a birth certificate is simply to search for the name of your state plus the phrase "birth records"; i.e., Illinois "birth records". You'll receive a huge variety of results that basically serve as placeholders pointing you to the state records offices; your best bet is to look for the URL with .gov or .us. These sites will have the information you're looking for either in an online archive or will tell you exactly what you need to do in order to track a copy down yourself. You could also do a search like this (using Google as your default search engine):
site:.gov "birth records" illinois
You'll be able to get county by county results using a search like this, which is obviously very helpful.
Some states store archival information via the state library system. In that case, you can try a search that looks like this:
birth records "state library" illinois
Now, this isn't as scientific a search query as the one given previously, but what this one will do is give you pointers to information on local sites that live and breathe genealogy (and are connected with the state archives/library in some way). You can narrow it by state URL as well:
birth records "state library" site:state.il.us
Start online, but be prepared to go offline too
The Web is a great tool for finding information, as we've seen in this article. Recent copies of birth records can be pointed to online, but in most cases, must be obtained either in writing or in person from the originating entity. Older records can be tracked down online using genealogical resources, such as FamilySearch.org. Either way, it's useful to know the different methods of tracing our family history.