JPEG is the simplest, smallest, probably offers best quality for someone who doesn't want to post-process the image themselves, but offers the least amount of control. When shooting in JPEG format, you can think of the computer inside the camera as being your own little photo-processing department. The camera will not only "capture" the image, it'll post process it by adjusting light levels, white balance, noise levels, sharpening, etc. It's all fully automatic.
HOWEVER... it will ALSO compress the image substantially so that it doesn't take much space. When it does this, it'll decide to reduce the quality of the image in areas where the human eye probably wont notice. This is lost data that you can never really recover. The "problem" here is that if something is over-saturated, too bright, or too dark, etc. a JPEG will just "compress" all the pixels that seem to be "alike" by making them effectively identical. When you later try to adjust the image by editing saturation, highlights, or shadows, you'll discover that the JPEG has clipped off all the detail and the area of the image is flat. This would not happen if you had shot in RAW.
RAW offers you the most control over your image. A RAW image is sort of like the "negative" in a film camera before it's been printed by the photo lab. The camera retains the original data as seen by the sensor at the time the image was captured and does not 2nd guess what the image was supposed to look like. It will make
some adjustments, but it will not make any adjustments that are considered "destructive" to the original data. The files will not be compressed -- so the storage of each image will consume much more space on your memory card. Areas of an image that are too bright, too dark, saturated, or blown out and "appear" to be flat, may not actually
be flat. You may actually be able to recover data by adjusting the photo -- this is because if too two pixels on the sensor appear to be
nearly identical but are not
actually identical, RAW will not compress them to be the same color (as JPEG would). It'll store the difference no matter how subtle. When you adjust the image, you may see the detail return.
HOWEVER... some things you may take for granted with a point-and-shoot will NOT be performed on a RAW image. For example, the camera will not attempt to de-noise the image. If you shoot in both RAW and JPEG then compare the results (without doing anything to either image) you should notice that the JPEG actually looks better. But that's because the JPEG was already processed and the RAW was not. After processing the RAW image, you may find that the RAW looks better. RAW is particularly better when you
know you'll need to adjusting the images.
TIFF is non-lossy (like RAW) but unlike RAW, TIFF is a fixed standard. E.g. if you need to give someone a non-compressed non-lossy file so they can do more processing with it then you send them TIFF. If you send them RAW you're taking a chance that they wont have any software that can open the file. The "problem" with RAW is that it's more of a concept than a standard. Every model camera seems to have it's own variation on 'RAW' and that's why there's a delay in support from Apple & Adobe to add 'RAW' support for any new model that comes out.