Photography's all about capturing light. The less of it you have, the longer you need to spend capturing it. This leads to blurry images as most things move and your hands will shake too.
You can partially solve this by:
Using more natural light - Shooting outdoors in daylight (can lead to harsh shadows and doesn't really work for your stated goal of shooting friends and family who tend to gather indoors for things like parties)
Supplying more light - using a flash (with the risk of redeye). Redeye is caused by light bouncing off the back of the eye on to the sensor. The closer the flash is to the sensor, the smaller the angles involved and the worse this problem gets. A flash hotshoe lets you move the flash away from the sensor. Also, external flashes tend to be angleable so you can bounce the light off ceilings and walls to get a smoother fill.
Reducing movement - You can put your camera on a tripod but it's a pain to carry around and a lot of compacts don't have mounts. You can also ask your subject to hold the pose but this annoys friends and most people other than trained models can't really do it. You also lose all action/candid shots.
Using a larger sensor - A larger sensor gives you a larger area to collect light.
Giving the light a larger hole to come through - Apperture. The problem is, the wider your apperture, the shorter your depth of field. A lot of compacts abuse apperture to make up for their small sensors but you end up with horribly shallow depths of field.
Amplify the signal - Rather than collect more light, you can amplify what you do get (higher sensitivy - ISO). The problem with this is photons hit relatively randomly with densities based on the light of the image. In large enough numbers (usually due to time), they average out and you get a nice smooth image. In small numbers, they're broadly but not exactly distributed based on the image you expect to capture. Amplify this noisy image and you get a lot of noise in the end result.
A DSLR solves most of these issues by giving you a much larger sensor than compacts use, uses higher quality components like microlenses, has much larger glass for collecting the image, provides a mount point for a better flash and gives you the ability to fine tune everything to get the right combination of tradeoffs for the shot you want. They also tend to come with much better autofocuses so you get the shot you wanted rather than wait for the focus to hunt and give you the shot a second after the action. For that reason, most people will suggest DSLRs - your odds of getting the shots you want are dramatically improved.
However - The best camera you can ever own is the one you have with you. If a DSLR is large enough that you never have it at parties, too expensive to risk at the beach, don't leave in the trunk of the car when out for road trips, it's completely useless except for the couple of times a year you plan a staged shoot.
Many of us with DSLRs realise and accept this so we see it for the tool it is, accept it may get damaged but a damaged and used camera is worth far more than an undamaged and unused one so we get a decent bag, toss it in the trunk, accept the weight of lugging it and all the glass everywhere and always have it with us. If you're like most normal people however, and won't do the above, a DSLR's a very expensive paperweight that's kept safely at home. Keep all of the information from the start of this post in mind and then find the compact with the fewest tradeoffs that's still small enough you'll have it everywhere (smaller size usually means more tradeoffs).
That might mean one of those credit card style totally flat cameras with a folding optic that goes everywhere. That might mean a basic compact with a zoom that comes out of the body. That may mean a larger compact with a larger fixed zoom. Or it may mean a DSLR. The point is, not knowing you and knowing what you will or won't put up with carrying, none of us can tell you what the right camera is for you. The best we can do is give you pointers to what will minimize your frustrations with a camera (namely ability to capture in non ideal light) and then leave you to decide what balance of size vs. tradeoffs is right for you.