What actually is the vector of angular momentum?

If an object spins around a central point, it gets angular momentum which is a vector with an orientation dependent on whether its clockwise rotation or anticlockwise, i get that. But what the vector part actually is, is confusing me. Does it mean that the object spinning gets a force going upwards or downwards? In that case, how do you explain bicycles? Shouldn't the spinning result in angular momentum pointing in one direction thus forcing your bike to fall over the same way someone would if they pushed you from either side of the bike?