There are three different ways the transition from "dating" to "relationship" goes down:
1. You're at a party with the guy and someone asks "is this your boyfriend?" You then look at each other awkwardly and have to decide at that moment if you're in a relationship.
2. One of you decides without telling the other one and then changes his or her Facebook status, much to the shock and surprise of all.
3. You both sit down to dinner and one of you asks the other if you'd like to make it official.
That third option is the old school way of doing things, but it's the way that I recommend. The best way to figure out if the guy you're dating is really your boyfriend is to talk about it. Ambiguity is the enemy of happiness, and conversation is the best way to get rid of ambiguity.
So talk it out. It'll probably be a very happy and exciting conversation.