First off, let me say that you can find the answer to this question in Sage using the nauty generator. If you're going to be a serious graph theory student, Sage could be very helpful.
count = 0
for g in graphs.nauty_geng("20 180:180"):
count = count+1
print count
The answer is 4613. But, this isn't easy to see without a computer program.
At this point, perhaps it would be good to start by thinking in terms of of the number of connected graphs with at most 10 edges. Then, all the graphs you are looking for will be unions of these. You should be able to figure out these smaller cases. If any are too hard for you, these are more likely to be in some table somewhere, so you can look them up.
Connected graphs of order n and k edges is:
n = 1, k = 0: 1
n = 2, k = 1: 1
n = 3, k = 2: 1
n = 3, k = 3: 1
n = 4, k = 3: 2
n = 4, k = 4: 2
n = 4, k = 5: 1
n = 4, k = 6: 1
n = 5, k = 4: 3
n = 5, k = 5: 5
n = 5, k = 6: 5
n = 5, k = 7: 4
n = 5, k = 8: 2
n = 5, k = 9: 1
n = 5, k = 10: 1
.
.
.
n = 10, k = 9: 106
n = 10, k = 10: 657
n = 11, k = 10: 235
I used Sage for the last 3, I admit. But, I do know that the Atlas of Graphs contains all of these except for the last one, on P7.
Yeah it's possible. For example take $n=6$ and each of the vertices have degree $3$. Then you can construct $K_{3,3}$ the complete bipartite graph on $6$ vertices and also another graph which isn't bipartite. The second graph can be obtained by constructing two distinct $C_3$ cycles and then connecting three pairs of vertices from distinct cycles. As both don't share a property (namely bipartivity) then they can't be isomorphic to each other.
You can find a picture of the two non-isomoprhic graphs here.
Best Answer
Maybe ask a few easier questions.
How many non-isomorphic graphs with 5 vertices and 3 edges contain $K_3$ as a subgraph?
How many non-isomorphic graphs with 5 vertices and 3 edges are connected?
How many non-isomorphic graphs with 5 vertices and 3 edges have more than 2 connected components?
I agree with the comments that suggest you should draw pictures, try this for smaller values, and explain what you have tried so far. I'm hesitant to give a more complete answer since this seems likely to be a homework question.