Compact or point and shoot cameras designed as the name suggests, for easy automatic shooting. No need to spend the time with many adjustments. The biggest weekness of most compact cameras is the small sensor (1/2.3″ which is roughly 4 x 6 mm ), which is usually very noisy in less then ideal conditions. To compare an entry level Dslr has a sensor of 22x15mm. This lets in 14x times more light. But nowadays more and more compact camera is equipped with bigger sensors as well.
There is an other category called MILC or CESC or Mirrorless cameras. These cameras has interchangeable lenses and usually bigger sensors. Most of them has the same size sensor as the entry Dslr cameras, and some of them has little small, but still much bigger than usual compacts. Still others has even Full frame sensors (36 x 24mm).
Today smartphones make better and better images. In fact some smartphones has bigger sensors than many compact cameras.
Currently for me Sony offers the most interesting point and shoot cameras. Their Rx line is the most reassuring.