Are you camping in the UK? There will be lots of great answers on this board but as a starting point:
(Also consider the size of your car before you start shopping. None of this has to be top-end expensive kit and if you look on eBay/marketplace you'll likely find lots of kit for sale.)
Something to sleep on/in. Airbeds are cheap, self-inflating matresses are more comfortable but bulkier and expensive, foam mats are cheap and might be ok for the kids for a night or two. Camp beds seem popular on here but obviously big to pack.
Something under your airbed/mat insulates you from the cold ground and helps to keep you warmer at night - foam mats or plastic/foiled bags picnic blankets. Some people like sleeping bags, others take duvets, pillows, blankets.
Something to sit on - Camping chairs and a table. Spending a weekend sitting on a picnic blanket isn't fun.
Eating Something to eat and something to cook it on! The briefcase stoves are ideal for simple meals. Take lots of snacks and think about things that you can just heat up. A cool box and some extra ice packs (check the site but many will have a freezer so you can freeze packs and then swap them each day). Plate, cups, cutlery, a sharp knife could be all be things that you have at home. A couple of pans.
If you'll have electric you need a special cable to run it to the tent. If you do have electric take a cheap kettle - much quicker in the morning for a cuppa.
Lighting Campsites get dark at night - stupidly I never thought about that the first time we went! Some lamps that you can have in the tent and then some torches for bathroom trips. Head torches are fun for the children.
Other random essentials that are in our camping kit:
Card games, a couple of puzzle books, something to keep the kids amused if they wake early, wellies/flip flips/crocs , boxes of wine, mozzie spray/citronella candles, suncream, bin bags, spare clothes that stay in the car so can't get wet, matches, washing up stuff including a couple of bowls, antibac wipes, kitchen roll, toilet roll, first aid kit...
I'm sure there's loads more. We camp regularly and I love buying camping kit and gadgets but you don't need a huge amount to get started.