What is this number?
The number next to that shield is your "achievement points". You earn them based on the point values of various achievements you have completed. These achievements include the ones from single player as well as multiplayer. Go to the achievements tab of your profile for more information on the achievements you have and how to earns the ones you don't.
If you meant the number next to your username (which is not where the arrow appears to be pointing), that is your player identifier. It is used by battle.net to distinguish between players with the same player names.
These points are gained from doing certain tasks in the game, for example, "Find out what Matt Horner won playing poker." This involves talking to characters in game.
Achievement Points are further broken down into categories so that people who prefer different parts of the game an accumulate them, and not feel as though they have to play JUST online multi or JUST the campaign. These areas are as follows:
Additionally there is a category: "Feats of Strength." This is a hold over from World of Warcraft which usually indicates achievements which can only be earned for a certain time period or by certain people. Currently, most of the Feats of Strength should be available to all (though hot shot can be permanently missed), though there is no guarantee this will be true going forward. Additionally, Feats of Strength do not affect your Achievement point total.
Each of these achievements has a point value associated with it which adds to your total number of points.
You can see the full list of achievements here