Can bitcoins be exchanged by mailed check? I am wondering as I'm (obviously) a newbie to Bitcoins.  I'm not necessarily looking to do so, simply wondering how transactions can be made/what options there are for exchanges
Yes, this is theoretically possible, but in practice you will not do this, because an internet connection is necessary for preparing the payment on sender side and for verifying it on receiver side. Thus, if both parties have an internet connection anyways, it is likely that there is no reason for mailed information printed on paper.

However, if one would still want to do so, he can.

You can print the private key of a bitcoin address, which holds bitcoins, onto paper. For example you can print the data in hexadecimal representation or as a QR tag.
You can then later recover this bitcoin address from the printed information and use the bitcoins stored on it.

If someone wants to send bitcoins by mail, he would use this technique. He would first prepare a new bitcoin address, put the amount of bitcoin that he wants to pay onto this address, and print the private key.
Next, he would send the printed private key to the receiver.
The receiver would then take the information, import it into his bitcoin software, and finally, to make sure the sender does not anymore control the bitcoins, transfer all the bitcoins stored on this address to a new address, of which the sender does not have the private key.

The last step might seem unnecessary, because the receiver already controls the received bitcoins. However, the sender might also still have the private key and use it for transferring the sent bitcoins, thus emptying the address of which the receiver has the private key. To make sure, this can not happen anymore, the receiver moves the coins away from the possible access of the sender.

Another possibility is to put a physical coin into an envelope and send it. This is basically the same thing as above, becuase the physical coins are just another way of storing a private key of a bitcoin address, which holds some money. See https://www.casascius.com/ as an example for physical coins.