The planets need gravity to form, and the sun needs gravity to form. The universe as we know it wouldn't exist at all without gravity, and we wouldn't exist either. The earth and the moon pull on each other with the force of gravity. If the moon got too close to the earth, the earth's gravity would pull it into the earth, and there would be a big crash. Luckily for us, that won't happen.

Scientists think there was a big crash when the earth and moon were being formed, before the solar system was stable like it is now:

read it here

The planets, stars, and myriad other "conglomerations" of matter which make up our universe were able to form precisely because gravity does exist. Gravity is one of the four fundamental interactions (or forces) and is the one responsible for holding matter together on a large scale. (Of the others, the strong and weak nuclear forces do not act over large enough length scales and cancellation of electromagnetic interactions with opposite charges are some of the reasons dominates on the "planetary" scale.)

Without gravity, there would be no force acting over a large enough distance to keep all of the smaller particles together and we would be left with a bland, featureless universe.

Without gravity we could not be alive, because:
(1) all of our atmosphere would boil off into space, and
(2) the sun would explode, since there would no longer be anything holding it together (the sun is basically a gigantic hydrogen bomb that can't blow itself apart only because it is so massive and has so much gravity).

Gravity is one of the four fundamental forces of the universe - that is, the four most important things that hold our universe as we know it. So I’m sad to say that not only would there be no planets and no sun and probably no humans, the universe itself would be completely different!

If the strong, weak, and electromagnetic forces still existed and somehow massless atoms existed (massless because anything with mass has a gravitational force. Or I supposed we could try to imagine some other kind of atom which has mass but somehow does not have gravity) probably the universe would just be a big, flat soup of atoms.