BGP operates as a rumor mill. Convergence is the process of all of the rumors settling into a steady state. The rumors are of the form "I can reach this range of IP addresses by going through this path of networks." Networks will refuse to listen to rumors that have themselves in the path, as that would cause traffic to loop.
For each IP range described in the rumor table, each network is free to choose whichever rumor they like best among all they have heard, and send traffic for that range along the described path. Typically this is the shortest, but it doesn't have to be.
ISPs will pass on their favorite rumor for each range, adding themselves to the path of networks. (They must also withdraw the rumors if they become disconnected from their upstream source, or their upstream withdraws it.) Business like hosting providers won't pass on any rumors other than those they started, as no one involved wants them to be a path between the ISPs. (Most ISPs will generally restrict the kinds of rumors their non ISP peers can spread, usually in terms of what IP ranges the peer owns.)
Convergence in BGP is easy in the "good news" direction, and a clusterfuck in the "bad news" direction. When a new range is advertised, or the path is getting shorter, it is smooth sailing, as each network more or less just takes the new route as is and passes it on without hesitation. In the bad news direction, where either something is getting retracted entirely, or the path is going to get much longer, we get something called "path hunting."
As an example of path hunting: Lets say the old paths for a rumor were A-B-C and A-B-D, but C is also connected to D. (C and D spread rumors to each other, but the extended paths A-B-C-D and A-B-D-C are longer, thus not used yet.) A-B gets cut. B tells both C and D that it is withdrawing the rumor. Simultaneously D looks at the rumor A-B-C-D and C looks at the rumor A-B-D-C, and say "well I've got this slightly worse path lying around, might as well use it." Then they spread that rumor to their down streams not realizing that it is vulnerable to the same event that cost them the more direct route. (They have no idea why B withdraw the rumor from them.) The paths, especially when removing an IP range entirely, can get really crazy. (A lot of core internet infrastructure uses delays to prevent the same IP range from updating too often, which tamps down on the crazy path exploration and can actually speed things up in these cases.)