How the number is measured
Response time is the time a pixel takes to change, quoted in milliseconds. Monitors usually give it as gray to gray, or GtG, which times a pixel as it moves from one gray shade to another. The usual method, based on a VESA industry standard, times the change from the 10% to the 90% point of the pixel's response curve, so it doesn't wait for the last bit of the change to settle.
A screen has many gray to gray transitions, and one quoted figure is a summary of them. Most monitors marketed for gaming list 1 ms, and some high end ones list less. A headline number tells you which products are fast. It can't tell you how one panel handles every transition.
Response time against frame time
The number matters when you set it next to how long a frame stays up. A refresh lasts 16.67 ms at 60 Hz, 6.94 ms at 144 Hz, 4.17 ms at 240 Hz and 2.08 ms at 480 Hz.
A panel that takes 8 ms to settle is done well inside a 60 Hz refresh. At 144 Hz it's still changing when the next frame arrives, so a trail follows moving objects. A pixel should finish before the next frame replaces it. That's why a faster refresh rate asks more of the panel, not only of your graphics card.
Pixels aren't the only cause of blur
Slow pixels cause ghosting, a faded smear behind a moving object. A different kind of blur happens even with instant pixels. Most screens hold each frame still until the next one arrives, and your eye keeps moving across it. The smear forms on your retina. At 60 Hz the eye dwells on each position for about 17 ms, and the blur shrinks as that time shrinks.
OLED pixels respond far faster than LCD pixels. So on an OLED, nearly all of the blur you see comes from the hold time, which backlight strobing or black frame insertion can shorten by showing the image for a smaller part of each frame.
Checking your own screen
A browser can't measure response time, but it can show the result. The ghosting vs inverse ghosting guide moves high contrast blocks across black, gray and light bands so you can read the trails. Dark trails mean slow pixels. Bright halos mean the monitor's overdrive is pushing too hard.