Taylor most likely will be fined and/or suspended by the NFL. The league's Conduct Policy states --
"Any Covered Person convicted of or admitting to a criminal violation (including a plea to a lesser included offense; a plea of nolo contendere or no contest; or the acceptance of a diversionary program, deferred adjudication, disposition of supervision, or similar arrangement) will be subject to discipline as determined by the Commissioner. Such discipline may include a fine, suspension without pay and/or banishment from the League."
Ray Lewis was fined $250,000 for pleading no contest to obstruction of justice. Taylor is pleading no contest to assault and battery, which are considered crimes of violence. Those are considered the most serious crimes in the Conduct Policy --
"It will be considered conduct detrimental for Covered Persons to engage in (or to aid, abet or conspire to engage in or to incite) violent and/or criminal activity. Examples of such Prohibited Conduct include, without limitation: any crime involving the use or threat of physical violence to a person or persons; the use of a deadly weapon in the commission of a crime; possession or distribution of a weapon in violation of state or federal law; involvement in "hate crimes" or crimes of domestic violence; theft, larceny or other property crimes; sex offenses; racketeering; money laundering; obstruction of justice; resisting arrest; fraud; and violent or threatening conduct."
Previously, players who have pleaded no contest to those types of charges have been suspended for one game (see Mike Doss, Leon Searcy, Leonardo Carson, Derrick Rodgers, Dwayne Carswell and Wayne Hunter, among others), so that's what I'd expect the NFL to do (unless it considers the crimes to be more serious because guns were involved, although Taylor denied that he had one).