Distilled water has had almost all of its minerals removed, so it will tend to taste "flat". It's not necessarily the best thing to drink as it can leach some of the minerals out of your body, but a little won't hurt you.
Bottled water is generally just spring or lake water that's been filtered to remove bacteria and other microorganisms, and is also sometimes passed through charcoal to remove chloramines and heavy metals if they're present in the source. Unlike distilled, the minerals are left in it. In fact that's the very benefit claimed.
Drinking water from the tap is virtually the same as bottled water, but it's also chlorinated to prevent harmful pathogens from growing in it while it's in the pipe. It can also pick up off tastes from house plumbing (rust, etc.) Filtering it will remove the off tastes and chlorine and so make it more pleasant to drink.
Spring water is the same as tap or bottled water, (a "spring" technically is an underground artesian source) but without filtering or chlorine added. But it's not always healthy - many upstream sources are contaminated with giardia from runoff, which is why backpackers take purification kits.