Knitting in the round is the act of knitting an object or garment in one seamless connected piece. It is achieved using circular knitting needles or double-pointed knitting needles.
In order to knit in the round, you must cast on your required number of stitches as you normally would and then "join" the two ends together. This can be accomplished by slipping the first stitch from the left needle to the right needle, then slipping the second stitch on the right needle over and onto the left needle. You can also hold the needles so that the needle with the working yarn is in your left hand and then knit as you normally would.
It is vital that you make sure that all of the stitches are facing the same way before you join the yarn, unless you are trying to make a möbius scarf. In that case, it is vital that you make sure thet stitches are not facing the same way. (It is surprisingly hard to make sure that the stitches don't twist twice.) You especially do not want to end up with a twisted loop when you are knitting in the round to make the body of a sweater or something else that absolutely cannot be twisted.
You can tell where the beginning of a row is based on the location of the tail yarn leftover from your cast on, but it is still wise to use a stitch marker.
Knitting in the round means that the "right side" of the work is always facing you, so your approach to certain stitches will change. Stockinette stitch is accomplished by knitting every row, while garter stitch is accomplished by knitting and purling alternate rows.
Knitting in the round is the opposite of knitting flat; you can still make objects such as hats and sweaters that way but must sew the ends together when you're finished knitting.