I have two datasets, one of animal migration patterns (collected over the course of a couple years) that consists of many points on an x, y plane (latitude, longitude), and the other of ocean surface temperatures (also a set of lat, long points). What kind of neural network can be used, if any, to see some pattern, or correlate the two sets, so that I could alter the temperature data and have the network predict how the migration pattern changes? Is this a feasible use of an artificial neural network?
Try a common feed-forward neural network with one hidden layer and train it with error backpropagatin to find a coherence between your input (temperature values) and output (migration patterns). In order to train a neural network, you have to have pairs of input and output vecors, i.e. in your case, one temperature measurement position corresponds to one migration position. If you cannot assign single input-output pairs, you might do some preprocessing (for example, simply grouping your data into a grid on the x-/y- plane and taking the mean of each grid). The question to find a suitable number of neurons in the hidden layer is discussed here.
Note that there exist many machine learning techniques to learn relations between input and output data. Furthermore, there exists a notable variety of artificial neural networks in structure as well as in training algorithms. As your question explicitly demands a neural network, this answer tries to provide a rather straightforward standard method.
For pattern analysis (in order to understand and/or visualize the patterns/correlations) some statistical approaches might be more suitable, e.g. clustering the data with k-means/EM-algorithm.