60Hz refers to how quickly the monitor refreshes, and it's pretty much meaningless these days. It tells us nothing about the monitor's size or resolution.
1080p is a different standard. It means 1920 pixels wide by 1080 pixels high (basically high definition). You can run 1080p with a variety of refresh rates, including 24Hz, 50Hz, and 60Hz.
What you should be looking for is the native resolution of the monitor. Is it 1920x1080? or something less (like 1680x1050, or 1280x1024) or something more (2560x1600)?
If your objective is 1080p gaming, then you should be looking for a monitor which is 1920x1080.