There are essentially two kinds of jig - those which count the quick eigth notes in triplets (1,2,3 ....) and those which count the emphasised beats, 1 per triplet. In reality, only the former are true jigs.
The characteristic of the tunes is that they are in 6/8 time which is arranged as two groups of three eighth notes per bar and follows the pattern 1,2,3 2,2,3. This form is commonly known as the Double Jig.
A Single Jig is similar (6/8 time which is arranged as two groups of three eighth notes per bar) but often follows the pattern 1 and 2 2 and 2 i.e. a quarter note followed by an eighth note in each group. The Slide is often confused with a Single Jig and may be written in 6/8 or 12/8, either way, it is counted as 1 and 2 2 and 2 or 1 and 2 2 and 2 3 and 2 4 and 2. The single jig is more akin to a Hornpipe or a Fling. Examples of true Single Jigs are Hag at the Churn and Off She Goes
A Slip Jig is similar to a Double Jig except that it is in 9/8 time arranged as three groups of three eighth notes per bar and follows the pattern 1,2,3 2,2,3 3,2,3.