Response time in the context of monitors generally refers to how much time it takes for a pixel to change color. It is often used in the context of shifting color from black to white to black again. However, there are other ways of measuring it. For instance, gray to gray transition (GtG) measurements (often used for gaming monitors specifically) track how long it takes for a pixel to transition between one gray level to another (commonly 10% to 90%). GtG measurements are evaluated as an average across different transitions. This response time is faster than the traditional BWB response time and offers some insight into the most basic layer of pixel response time.
Sometimes "response time" is evaluated in terms of MPRT (Moving Picture Response Time). This is not a measurement of how fast a pixel changes color, but rather for how long a pixel is continuously visible. Even when a pixel finishes changing color, it can remain in that same state until the next time the screen refreshes. It's commonly used when trying to represent visible motion blur caused by this persistence. It is often limited by refresh rate, but specific blur reduction technology can often lower it further. Some monitors offer an "MPRT" mode which signifies how the blur reduction technology is applied. In many instances, the mode it is set to can affect other monitor performance characteristics, such as brightness.
Modern LCD displays usually exhibit a response time under 10ms (for B-W-B transitions). Lower is better here, as it cuts down on some common image issues like "ghosting" and blurring. GtG response times usually measure under 5ms, with higher end monitors offering close-to or even sub-1ms GtG.