I'll start with VGA, which is, in my opinion, a "standard".
VGA is an analog signal. This basically means that the quality will not be as great as cables that make use of digital signals, especially at high resolutions. The problem with analog is that noise often distorts the signal and the quality ends up not being as good. So in terms of quality, go for the other three.
Digital signals do not have this problem because they are noise tolerant.
DVI is a digital signal that has a few different connecter types and two link modes:
- DVI-A has an analog signal.
- DVI-D has only a digital signal.
- DVI-I has both, making it especially useful as it can be used with VGA-DVI adaptors
Most adaptors are single link and support a resolution up to 1920x1200 whereas dual link supports up to 2560x1600.
HDMI is the standard for HDTVs. It is also a digital signal and is therefore fully compatible with DVI-D and DVI-I. The benifit of HDMI over DVI is that is has the ability to carry audio as well as the video signal although it does require that your monitor has built in speakers. Recently, HDMI 2.0 was released and gives HDMI the ability to better deal with 4K (Ultra HD) displays at a higher frequency and FPS. If your display or TV has HDMI 1.4, you’ll be limited to 3,820×2160, 4K, at 30 Hz. However, if you’ve got a video card and 4K display with HDMI 2.0, you’ll be able to get 4K at 60 Hz.
There's not much difference between between HDMI 1.4 and DVI besides the audio in the HDMI, which can be a real benefit depending on your setup and that colour ranges for HDMI go beyond the RGB spectrum which DVI is limited to. The only benefit of the DVI is the physical screws that support the DVI cable and help prevent port damage.
DisplayPort is the newest of the lot and is aimed towards higher end monitors - newer and high-end monitors tend to have DisplayPort. Its designed to deal with 4K at higher FPS and Hz. DisplayPort is capable of 3840×2160, 8K at 60Hz or 4k at 120Hz!
DisplayPort looks like it's the way of the future, especially if you want 4K.
Generally it depends on your setup (monitor and graphics card), what you can and can't use but for the future of gaming DisplayPort looks the way to go.
References and more info: