How to check a motherboard, is it in good condition or not?. There is no display from motherboard?

Answer 1

Remove all other cards and drives from the board, even the memory, turn it on, you should hear a series of beeps, but it will depend of the motherboard brand.

Answer 2

Unless a visual inspection of the board reveals an obvious problem [leaking capacitor, broken circuit path or component, etc.], the only way you can check it is by a process of elimination.

Since you have no display, you need to check the monitor, the cable connecting the video card or chip to the monitor and the video card or chip itself.

If you have another computer that is working, connect the monitor and cable to it. If you get a display, then you have eliminated those two items.
{If you don't get a display, either the monitor or cable is bad.}
That would leave the card or chip itself.
If you are using a video card, try connecting the monitor cable to the onboard video chip after removing the card from the computer. If you get a display, then the card was apparently bad.
Conversely, if you are using the onboard chip put a video card in the system. If you get a display, then the chip is bad.

If you don't get a display either way, the board could still possibly be good. Failing to display due to a short due to a bad installation of the board [some component contacting the case somewhere] or a bios problem.
The standard technique to correct bios problems is to remove the battery for at least one minute (with power off), then replace it. This resets the bios, which will generally reset the video entry to accept whatever device is present in the system.