Like in any artillery game, the objective of the game is to blast the opponent into smithereens. In Worms each team has four worms and using an arsenal of weapons, ranging from grenades and bazookas to shotguns and explosive sheep, you have to eradicate the other team. Part of what made Worms such a good game was this wide variety in weapons to choose from, and the fact that each weapon was actually useful in different situations. Seasoned Worms players, who were able to throw a grenade with immaculate precision and timing, could bounce grenades off of walls and hit you even when you thought you were very well hidden. The action is divided into 60 second turns, where you have to position your current worm correctly for the shot, shoot, and retreat to a safer location. To add to the fun, chain reactions were a very common thing in Worms - you'd shoot at a worm, who'd then fly into a mine that bounced onto another worm, who then exploded (worms explode when they die) and set fire to a barrel of gasoline, that ran down a hill and pushed a third worm into the water. Setting off these chain reactions is immensely fun, whether they are planned or not :-)
Not all of your items are weapons. Some are used to either defend yourself by building barricades, or to help you move around the level. Expert players soon learned to use these accessories - e.g. the excellent ninja rope - to move quickly around the level, place a stick of dynamite, and then make a hasty retreat.
Fun is definitely the keyword with Worms. This was not a game that took itself too seriously. The game included speech, and the worms all had little sentences that they would say during combat. Like saying "watch this" or "fire" when firing, muttering "revenge" when hit, saying "bye bye" when dying and so on. All of these were done in a very cute little voice, which made it all the more adorable amongst the carnage that is Worms.
Worms turned into a series of games that is still running to this day. I fondly remember Worms Armageddon (1999) and Worms World Party (2001), but there has been countless other entries in the series, the latest one being Worms Battlegrounds which was released for the PS4 and XBox One this year.
The original Worms is an excellent game so I'd definitely recommend that you give it a go. I'm not sure where you can find the original game, but the second game in the series, simply named Worms 2, is available at gog.com.