As the WP page says, there are two distinct meanings of "swing": groove or feel, and playing unequal notes. I say distinct, but of course the latter feeds into the former.
A common example of unequal notes is the hornpipe. These tunes are customarily written down with pairs of equal quavers, but that is not how they are played. You can think about the swing in different ways, depending how your brain works:
- The first quaver is lengthened and the second quaver is shortened;
- The second quaver is played late;
- It is played as a crochet + quaver triplet, or as a dotted quaver plus semiquaver, or somewhere in between.
Another common example is the Viennese waltz, where the second beat of the bar comes fractionally early, and the third beat slightly late (though I don't think this is usually described as swing).
Lilt is similar, but (to me at least) lilt refers to triple-time tunes and swing to simple time. But I may have just made that up.
As for playing with swing: the ideal is to "just know" how it feels, and have sufficient command of your instrument that you can make it happen. For the rest of us... I dunno. If you want to analyse it, you could try getting a computer to play the tune, which it will do absolutely soullessly and straight, and compare that to the concertina in your head (which, if you're anything like me, sounds much better than the concertina in your hands) and see what you would emphasise differently. Emphasis can be playing a note a bit louder (or suddenly softer), or a bit more (or less) staccato, or for a longer duration, or with an appoggiatura, or whatever other musical effect you can think of. Different styles of music will prefer different effects, of course.