What happens if you plug a 1080p monitor in to a 720p laptop computer? Will it work/look good?

In a double screen arrangement with expanded presentation the PC's OS will change the goal of the outside screen to 1080p and keep up the implicit showcase to 720p so no issue.

Yes. The monitor will automatically support lower resolutions than its screen is capable of... and the laptop probably has external graphics better than its own screen. Look up the specs for each, you may be pleasantly surprised.

Depending on your laptop's GPU, you might be able to connect to higher resolution monitor. Your laptop comes with built in 720p monitor not because it's the limit, but because that's what you are willing to pay more, the same model laptop might offer higher resolution display, for an additional fee.

You could identify your laptop GPU (graphic card) by going to dxdiag (Win+R to open run command, then type dxdiag, enter). On display tab you will find everything about your graphics card, it should show either Nvidia/AMD if your laptop comes with discrete GPU, or Intel HD if it doesn't. Google your GPU to find out it's maximum output resolution before buying a monitor with higer resolution than your current laptop display. Or just plug it in and find out if you already have the monitor lying around.

yes it will if its compatable ..

Why don't you just do it and find out.

You are not going to blow anything up.

You just wont get 1080p performance.

Yes, Your laptop monitor doesn't affect the performance.

In a dual monitor setup with extended display the laptop's OS will adjust the resolution of the external monitor to 1080p and maintain the built-in display to 720p so no problem.

it will run at 720p and look like a 720p monitor

No matter how good your monitor is you will only get the resolution that your computer is capable of transmitting to the monitor

It’ll look like it’s 720p.