Does dropout require multiple passes of the same data set, as a sort of ensemble method?
Dropout is applied over one network. Sometimes (like with non-dropout networks) you will run your data through it multiple times before it converged and this number will be a bit higher on average with dropout but it is one network. Per layer you have a dropout probability during training and during testing/prediction you use the full network without dropout by multiplying the weights by (1-p) where p is the dropout probability. This is because during training (1-p) of the nodes are actually used.

Why this is related to ensembles is that every training instance is basically trained on a different network, by randomly dropping out nodes, forcing it to learn specific things using different nodes because it will not have all the nodes available at all times. It is not a traditional ensemble in that you combine multiple networks, just during training it acts a bit like it.