Higher resolution is always better right now, as the monitor is most likely close to you and you will be able to differ from the different resolutions, so aim for 4K right now, as 8K monitors do not exist right now. Do not get a 720p monitor by all means.
A very important aspect of choosing a new monitor is the maximum supported resolution (native resolution). The resolution is basically the amount of pixels in the width and the height. A 1920×1080 resolution means that you have the width consists of 1920 pixels and the height consists of 1080 pixels, totaling 1920 * 1080 = 2073600 pixels. The most popular resolutions are:
1280×720 (HD, 720p)
1920×1080 (FHD, Full HD, 2K 1080p)
2560×1440 (QHD, WQHD, Quad HD, 1440p)
3840×2160 (UHD, Ultra HD, 4K, 2160p)
7680×4320 (FUHD, Full Ultra HD, 8K, 4320p)
Okay, you might not find an 8K monitor anywhere, but hopefully we will see them around in 5-10 years.
Which resolution is best?
Higher resolutions means more pixels to get you a clearer and better image, and since we are in 2015, you shouldn’t get a 720p ever. Not even a used one. 1080p is fine for gaming purposes because you can hit 144FPS easier (if you have a 144Hz monitor of course), but falls flat on all other purposes. If you are looking for a monitor for console gaming only (PS4, XBOX One, etc.) then 1080p is also the best choice, because you can’t get higher resolutions in any games anyways since the consoles do not support it. They currently also support up to 60Hz refresh rate only. 1440p seems to be the sweet spot between resolution and frame rate, as you can get 144Hz refresh rate too and it is easier to hit the desired 144FPS with current-gen graphics cards. 4K is currently the best resolution you can get on a computer screen, but if you are on a budget, you might consider a lower resolution. A budget 4K monitor, such as the Samsung U24E590D which costs $399, which is more than most budget builders are willing to spend.
So generally speaking, higher resolutions are always better. The only things you have to worry about are limitations that the panel brings and the added price. For example, you can not get a 4K monitor with 144Hz refresh rate, but you can get a WQHD monitor with one. That’s one important thing to consider, so you have to prioritize what you need in a monitor the most.
Refer to the image below, for example. It’s for TV, but it also applies for monitors. Since your viewing distance is short because it’s a monitor and not a TV, you can see that higher resolutions are always worth it and even higher than 4K (Ultra HD). 8K anyone? Asus? Samsung? LG? No? Okay then.
For entertainment purposes only, we found this video useful and at the same time very entertaining. Nothing against consoles, by the way.