I understand that the "degree" of a vertex of a graph is equal to the number of edges present in a vertex. So I am assuming that the "degree sequence" of a graph is the sequence of degrees for all the vertices. Is that correct?
Additionally, what does it mean to say "uniform distributions over a specific degree sequence" (in the definition of configuration models, for example)?
Yes, the degree sequence is simply a list of the degrees of individual vertices. When people talk about a uniform distribution over a specific degree sequence, for example choosing a random three-regular graph on a given vertex set (where the degree sequence is $3,3,3,\ldots$), what they mean is you want to consider all possible graphs with that degree sequence and choose one at random, each with probability $1/M$ where $M$ is the number of such graphs.
Of course this would be very difficult to do if we needed to list all possibilities and then choose one at random; even calculating $M$ is hard. However, the configuration model is an easier way to choose one uniformly at random without doing this: generate graphs using the configuration model until it produces a simple graph, and then keep that one. Every labelled simple graph with the right degree sequence is equally likely to be produced.