I may be incorrect, but I was taught that a Dial Indicator (DI) generally reads .001" per graduation, and has 1/4" or more of movement. Dial Test Indicators (DTI) generally measure .0005 or less per graduation, and have limited measuring range.
DI's are used for "fairly accurate" work, whereas DTI's are used for really accurate work. When centering work in the 4 jaw chuck that needs to be dead-nuts accurate, I'll start with the DI (.001" graduations), and get the runout (needle movement) as small as possible. Then I switch to the DTI (.0005" graduations), and refine the centering of the workpiece until it shows as small a movement as possible.
DI's are available in digital flavors, but I dislike them personally. I prefer mechanical DI's with a needle, because it shows more than increments of .0005".
As for which ones to get? I use cheapie indicators from Harbor Freight. They're inexpensive (Around $14-15 or so) for a mechanical DI, and I have 3 or 4 of those. I bought a DTI from Little Machine Shop, 0-15-0 (max deviation of .03" on either side of zero), with .0005" graduations. I think it was around $30 or so. It's not a Starrett, but it works for me and has proven accurate enough for my shop.
I have an old Federal DTI that I need to send off for repair, but the hobby fund is depleted.
I highly recommend a Noga type indicator holder, with the single locking knob. I bought one on sale about a month ago, and I love it. They're around $40 or so at LMS, and worth every penny. I've no complaints about mine. Beats the heck out of the non-articulated type that have two locking knobs. Indicators and magnetic base indicator holders are available all over the internet, and they're useful for far more than just holding an indicator. I mount my camcorder on them, my Nikon DSLR, use them to position chip deflectors when fly cutting, etc, etc. Handy little things.
Also, if you have a lathe, one of the inexpensive back-mounting magnets for a DI is a great thing. Sticks to the ways, you position the carriage, dial the indicator to zero, and you can return to the same point repeatedly. I prefer it to the depth stop on my lathe, unless I'm making multiples of an item. You can power feed against the indicator without crashing the machine (provided you don't crash the tool into the shoulder), but power feeding into a hard carriage stop gets expensive in a hurry.
Just my $.02