Connections are for linking two or more two skills of medium or high difficulty in a row. You must not pause or wobble between skills!
There are two bonuses in FIG (this might not be the same where you are), 0.1 and 0.2. You can also be deducted 0.1 for not connecting. I can't think of many examples but I've got some.
A D with flight from high bar to low bar connected to a C will give you O.1 (although they have to be performed in that order), and so will two Ds.
C+D elements must have flight or at least a half turn to get the bonus.
To get O.2 you can use a D with flight, on same bar or to other bar, and a C on the high bar, again in that order.
Watch Ksenia Semenova's 2008 bars for her tkatchev/ giant full, this would be a 0.2.
An element can be performed twice to get a bonus, but won't be counted again difficulty wise. You can also use your dismount as a connection, but any empty swings will lose you the bonus.
There are still 0.1 and 0.2 bonuses, but three different categories.
Acrobatic: 2 acro flight elements, but not the dismount/mount.
For 0.1, you could have:
C/D + D, for example front tuck back tuck,
C + C,
B + E.
These two would include the mount and dismount.
B + B + D, Bhs, bhs double tuck.
C + B + C
B+ F, and C/D + D, but there's a special condition attached to getting this one I don't quite understand.
Including mount and dismount:
B + B + E, (bhs bhs double pike, 2 x bhs + layout (done by Shawn J))
C + C + C, (switch split leap, LOSO, back pike salto, again Shawn J)
B + C + D
A+ C or C+ A.
C+ C, excluding dismount.
D (salto to one foot) + A (scale)- eg Nastia/ Steliana Nistor's original moves.