I'll assume your graph is undirected. (Set NodeXL, Graph, Type to Undirected.) The answer is a bit more complicated in the directed case. To find out how many unique vertices a given vertex is connected to, do this:
1. Select NodeXL, Analysis, Graph Metrics.
2. Check Degree in the Graph Metrics dialog box and click Calculate Metrics.
This will add a set of Graph Metrics columns to the Vertices worksheet. The number you want is in the Degree column.
To see the connected components in a graph, do this:
1. Select NodeXL, Analysis, Groups, Group by Connected Component.
2. Refresh the graph.
The vertices in each connected component will be given a unique color/shape combination when you do this.
There are many other things you can do with groups, such as displaying them in separate boxes, collapsing them, and getting metrics about them. Go to NodeXL, Help, Help and see the Working with Groups topic for more information.