What happens if my spouse wants a divorce and I don't?

Well, in New York, it means you make him/her prove that they are entitled to a divorce. In New York you have to prove grounds to get a divorce. You have to provide the court with a basis to say that one or the other party violated the marital contract – because, marriage, as you may remember, is a contract.
So you can answer your spouse’s complaint and deny all the allegations in the summons and verified complaint. If you play hardball, and your spouse is unable to prove that you did anything to them that would warrant a nullifiction of the marital contract, then you win. There is no divorce. At least not legally.
The problem, though, is that you would be married to someone who no longer wants to be married to you. That could be very demeaning. Your spouse could emotionally or physically abandon you and there is nothing you can do to stop that. Plus, eventually, your self esteem and self respect are bound to take a nose dive. Obviously people have their reasons – religious, cultural, personal – why they would object to a divorce.
But normally, by the time someone asks for a divorce, the marriage is over. Hanging on to it can only become emotionally harmful.
One reason someone might hang on is for leverage. The other side may be willing to pay more in terms of equitable distribution, cash, property, maybe even agree to certain custodial scenarios to get out of the divorce. So that is a reason to object to the divorce. But if it is just a question of, “I love my husband/wife and I won’t give them a divorce even though they want it,” you could win the battle, but you really won’t have a marriage.
So it is usually better to concede.