The honest answer: probably not, unless you play with other people
Most "do you need a second monitor" articles conclude that you do, which is convenient for everyone selling monitors. The real answer depends entirely on how you play, and there is a clean line through it.
If you play alone, mostly single-player, you probably do not need one. A second screen does not make games look better, run faster or feel more immersive. It will sit there dark for most of a single-player session. The money is better spent on the main monitor, the graphics card, or games.
If you play with friends, the case is much stronger. Group play generates a constant stream of things that want to be visible at the same time as the game: voice chat, who is in the party, someone sharing a screen, the text channel, the guide for whatever you are all stuck on. On one monitor every one of those is an alt-tab out of a session other people are waiting in. That is the problem a second screen actually solves, and it is a real one.
Four questions that settle it
Do you currently alt-tab during sessions? If you genuinely do not - if you play with voice on your phone, or you play alone, or you simply do not look things up - then a second screen will not change your evenings. If you tab out several times an hour and it annoys you or your group, that is the whole argument.
Is your desk big enough? Two monitors on their factory stands take an enormous footprint, and the stands prevent you from pushing the panels together. If the desk is small, the honest budget is a monitor plus an arm, not a monitor - see our arm guide for what to check before buying one.
Does your graphics card have a free output? The most common reason a new second monitor ends up back in its box. Look at the physical ports on the card with your current monitor plugged in and see what is actually free, then buy a panel with that input.
Do you already own a monitor you could demote? If you were going to upgrade your main display anyway, moving the old one to the side is the cheapest possible version of this and costs nothing extra beyond a cable and possibly an arm. Different panels work fine side by side for this purpose - mismatched monitors covers the specifics.
What to buy if the answer is yes
The single most useful thing to understand is that a secondary screen is judged on different criteria from a main one. Refresh rate barely matters, because a chat window does not benefit from 165Hz. Response time does not matter at all. Size matters less than people expect - a 24 inch panel holds a voice client and a browser tab comfortably, and a larger one mostly takes desk space you needed.
What does matter is viewing angle, and it is worth being specific about why. A second monitor is angled toward you from the side, so you are always looking at it off-axis. IPS panels hold their color and contrast at an angle; cheaper VA panels shift noticeably. That is the one specification worth paying a little more for here, and it is why the IPS options rank above the cheaper VA one below even though the VA panel saves real money.
The second thing worth having is the right input. A monitor with both DisplayPort and HDMI will connect to whatever your card has free; a monitor with only HDMI and VGA might not. And the third, if the second screen will hold your voice chat, is built-in speakers - not because they sound good, they do not, but because pushing Discord audio to them lets your headset carry only game sound, which is a genuinely nicer way to play.