Charged with battery not sure how to plea, and what is no contest plea?

In order to determine how to plead, you really need to discuss your case with an attorney who practices criminal law in the same area where you are going to court.  A local attorney will know the prosecutors, the judge and how these types of cases are normally settled in court.  I am not licensed to practice in New Mexico, but in California.  I can tell you generally that a "no contest" or "nolo contendere" plea will result in a criminal conviction.  It just means that you are choosing not to contest the charges.  Sometimes, defendants plead no contest instead of guilty so that there is no admission that can be used against them in the event the "victim" sues them in civil court.  You need to consult a New Mexico attorney.