You are smart to be suspicious. It could be a nice thing or a huge problem. So many people do not ask questions, they just think "Yes!!!" and go cash mystery checks. Weeks later, the felony charges drop on them and they are upset and confused and, oh yes, they have already spent the money. It's far better to find out what is going on beforehand.
I would suggest first contacting the bank on which the check is allegedly drawn to find out if the check is valid and if there is sufficient money in the account to cover the check. Tell the bank the circumstances under which you received the check. Perhaps it is all fine and the bank can assure you that the company is reputable and valid. Perhaps, alternatively, they can let you know if there is a problem. The problems could include that there is no such bank account, that this bank account does not belong to the company on the check, or a long list of other possibilities.
Cashing a fake/forged check ... you don't want to do that. It is very hard to explain why anyone else, unknown to you, would send you a fake check. The only one who could benefit from such a scam is you, and only for a little while. The assumption, therefore, is that you knew perfectly well it was fake/forged, and you made it yourself. It looks bad.
If the company is reputable, you should be able to contact them for further information as to why they sent you a check. Do that. Ask for a letter on their stationary explaining why they sent you a check. If everything makes sense, and you have done your homework on the mystery check, and the bank says the check is valid, then you can go cash it.
Alternatively, you could be reading the whole thing wrong. It could be an advertising check -- not a real check. Some sweepstakes "you may have won $XXXXX!" checks look awfully real. A bank won't let you cash one of those (hopefully.)
DISCLAIMER I do not practice law in your state. This answer is provided solely for general informational purposes only. This answer does not constitute legal advice, create an attorney-client relationship, or constitute attorney advertising
If you've never heard of the company, then chances are good that this is a scam. You haven't been contacted by email by anyone from another country trying to hire you and advising you that they'd be sending you a check, some amount of which you get to keep, after you immediately send them the balance? if this check is drawn on a foreign bank, if it's a forgery, it can take 2-3 weeks to bounce -- long after you've written your own check to someone else.
You can contact the bank to see if there's actually an account in the accountholder's name there, but they won't tell you if there are sufficicent funds or not, and they can't tell if the check is a forgery until they try to process it.
You can also report this to the FBI, linked below.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Check in the mail scam - but you could discuss it with your bank and see if they are willing to cash and out a hold onto it and then you should wait several months to see wha develops if you think the check might be legit but usually they're not and will damage your standing with the bank.