DVI is better, it's necessary for high resolutions, also VGA is built into DVI, meaning you can run VGA out of a DVI port as long as you have a plug converter (signal compatible, physical cables aren't).
Wow, that was wildly inaccurate.
"VGA" (DB-15) is still a 'digital' signal, but being fed over an analog connection. Because of this, it can have signal integrity issues in electrically noisy environments, or if cheap DAC/ADC components or cables are used at high resolutions. It caps out around 2048x1536@60Hz in terms of bandwidth, and towards the high end of that many cheap video cards may have quality issues.
DVI is a totally digital spec that uses differential signalling (like Ethernet) to make it much more immune to electromagnetic interference.
Single-link DVI is good up to 1600x1200@60Hz -- and can usually be pushed to 1920x1080p@60Hz, but this is technically out of spec and requires things like reduced blanking interval.
Dual-link DVI is basically two DVI connections running in parallel to feed one display, and doubles the effective resolution. This is needed for those 2560x1600 30" LCDs, and other specialized high-res LCDs (for medical imaging, etc.) Dual-link DVI
is not the same as having two single-link DVI ports on one card; each individual port is either single- or dual-link.
As part of the DVI spec, you can use some of the extra pins in the DVI port to run a VGA signal. This lets you output either DVI or VGA from one port -- but you are not actually converting or transcoding anything, just moving some pins around.
There are three kinds of DVI ports:
DVI-D: DVI (digital output) only. You can't get VGA at all out of a DVI-D port.
DVI-A (rare): VGA (analog output) only. An adapter cable or plug must be used to use this port at all. But usually something that outputs VGA only will use a DB-15 plug (the 15-pin "VGA connector").
DVI-I: DVI or VGA output (usually one or the other, not both at once). An adapter cable or plug must be used to get at the VGA output. This is what most -- but not all -- "DVI" video cards have, enabling them to drive either new digital displays or older VGA monitors.
HDMI is DVI bundled with digital audio in a single cable (for home theater, so you only have to run one cable from your receiver or cable box to the TV). You can convert between them (the video part, at least) with a simple adapter.