You do not have a tricep. You also do not have two triceps, one on each side. What you do have his triceps (plural, not singular) on both arms. The important thing to remember here is tri. Three.
There are three heads to your triceps. Tim Henriques wrote an article for allaboutpowerlifting.com that gives a brief overview of this anatomy and shows all three triceps heads.
Your triceps has 3 heads – the Long Head, Lateral Head, and Medial Head (feel free to stop listening to someone the instant they start talking about the non-existent ‘short’ head of the triceps). The Long Head is unique in that it is a two joint muscle – it crosses both the shoulder and the elbow. The Lateral Head is on the outside of the arm and in my opinion it is the easiest to develop with training (basic pushdowns will do the job). Unfortunately it is also the least important when it comes to bench pressing. The Medial Head is the smallest head, it is on the inside of the arm above the elbow. You would generally only see this when you bring your arm up and flex your bicep. See the pics below for more info.
If you're interested in how each of these three heads works in building a bigger press, Henriques covers that, too:
All of the tricep heads produce elbow extension, which is just straightening out your elbow as in a pushdown or a bench press. Only the long head also produces shoulder extension which is when you bring your arm down in front of you (think of a Karate chop). The elbow needs to be relatively straight for the triceps contribution to be significant to this motion. The long head is important in maximal strength because it crosses the shoulder joint and adds significantly to joint stability – a bigger long head means more tissue to compact the joint which means more force can cross the joint (see the book All About Powerlifting for more detail about the importance of joint stability and how that relates to strength). It is extremely rare to see a really good bench presser without a very well developed long head. The medial head helps perform the lockout portion of elbow extension, it contributes to those final inches of a press. It is also pretty important in bench pressing. The lateral head also helps out but in my opinion, because of its angle of pull, it is the least significant contributor of the 3 heads in the bench press.
This may not tell you everything you need to know about your triceps, but it's a good start.
Image and text via allaboutpowerlifting.com/killer-triceps/