Does it tell you why it wants you to press F1? I've seen this for several reasons, and it is usually because the system self-check failed. If it says something like, "CMOS Battery Low Voltage", "Checksum invalid", or "System time not set" (these are the ones I see the most), then the battery on the motherboard that keeps the clock and BIOS settings is dying. Usually this is replaceable, and if the battery is the cause, you have to replace it to get rid of the error.
If you think it's the battery, just open the case, look for the battery (should be roughly the size of a US Quarter), write down the battery type (probably CR2032 or similar), and buy a new one. They should have them at most electronics stores or anyplace they sell watch batteries. Depending on where you buy it, it might be around $3 USD (no more than $5 for a single battery). Unplug the computer, make sure the battery is not soldered in (if it's in a plastic clip then it isn't), and pop it out with a small flat-head screwdriver, then pop the new one in the same way the old one was (most likely + side up). Plug in the computer and boot it up. You will see the error (or a similar one) one last time. Press the key to go into setup, then go to the exit screen and select the option for exit and save changes. That should solve the issue.
If it says something about a fan or sensor, then you definitely want to fix it, because if it's a fan issue, the computer could overheat much more easily. If it's a sensor issue, the computer would not be able to detect an overheat condition, and if it overheats, it won't shut off in time to prevent permanent damage. In these cases, open the case, and make sure there are no loose connections. If you see a cable hanging around disconnected and don't know where it goes, leave it alone (there will be a few of these coming from the power supply). If you find one that goes to a fan or a little bulb-shaped sensor, and can't figure out exactly where it goes, you should be able to find a diagram of your motherboard online that will show you where the cables should go.
If it says anything about the HDD (S.M.A.R.T. test failure or warning), make sure you have proper backups and start shopping for new hard drives.
If it says something about the RAM and you haven't added or removed any recently, you might need to replace your RAM. If you recently added or replaced RAM, make sure it is installed correctly.