No contest simply means you are not contesting the charges (you are not admitting guilt, but you are not going to fight the charges and will accept a sentence).
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I agree with the other attorney, you are not admitting guilt, you are simply stating you do not want to contest the charges anymore. Unless you were speeding significantly you should be fine. If you are still unsure, you should hire an experienced ticket attorney to help with your case.
Entering a no contest plea does not automatically mean you will not get points on your driver's license record. To avoid points request that the judge or hearing officer withholds adjudication of guilt.
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Pleading no contest will not mean that you do not get a point, it is as my colleagues have said not admitting guilt, but it is treated the same way as a guilty plea by criminal courts. You can plead no contest and ask for traffic school, that will keep you from getting a point on your driving record.
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I would recommend speaking to an experienced traffic lawyer before you enter any plea. As stated by other responses, stating no contest does not ensure you will not receive points or other sanctions I'm sure you are hoping to avoid. There may be legal defenses to your ticket that you could explore in court. Additionally, depending on the speed of the ticket you may be facing additional consquences outside of just a fine and points. Before you head to court speak with an experienced lawyer.