Bitdepth
Bitdepth is one of the most important considerations when choosing a budget display. Fortunately, the rule is simple: never buy anything with less than a 10bit panel.
To explain the difference extremely briefly, 8bit means a video codec or display is only capable of 16.7 million colors. That sounds like a lot until you compare it to the 1.07 billion colors a 10bit panel (or codec) is capable of. It's quickly apparent why a 10bit panel is so superior to 8bit. This is especially true when working in codecs that are 10bit (or greater). An 8bit monitor would be incapable of displaying all the colors!
A result of the fewer colors can be the introduction of banding in your images. See the below image for an example:
Compare the 8bit images (top) with their 10bit counterparts (bottom). If you look closely, you can see the tiny bars that appear in the 8bit palettes. These are introduced because there aren't enough colors to accurately display the whole spectrum. Compare this to the smooth gradient the 10bit images display with their larger amount of colors. This is a simulated comparison, but fairly accurate as to what you might see when comparing an 8bit/10bit display.
There are some monitors which boast about their "10bit processing." Generally, these are using clever electronics to mask banding and other 8bit issues while still using an 8bit panel. This does not overcome the color difference. If a monitor has a 10bit panel, the manufacturer is going to proudly display that fact.
Equally important to having a display capable of 10bit is ensuring your connection and video card are capable of sending a 10bit signal. If a single part of the chain is only capable of 8bit, then the output will be 8bit.
DVI is (generally) not capable of 10bit (though a few monitors have an implementation), DisplayPort and HDMI are. NVIDIA GeForce video cards, while able to output 10bit in DirectX mode, are effectively 8bit in Premiere, etc falling back to OpenGL rendering. NVIDIA Quadro cards are 10bit because of their 10bit color buffers. AMD is similar with their Radeon cards only outputting 8bit. Their proline FirePro and FireGL cards are a must to achieve full 10bit capability.