[Technique] Can someone teach me how to round out edges?

An example image in the URL, a lot of the lines are very obviously chunks of straight lines, when they're supposed to be smooth and round.

Not really sure what's the term I should be googling to learn how to deal with this. Anti-alias and "round edges" don't seem to be right.

I'd greatly appreciate either directly rounding out this image for me, or just telling me what terms to search for to learn to do it myself! Thank you!

EDIT: Oops. Seems I didn't attach the URL properly. Here it is! http://img11.deviantart.net/42b4/i/2010/248/3/f/sos_brigade_logo__hd__by_suchipi-d2y495n.png