The correct answer has been posted, but this was my initial guess.
I think you are:
A ring buffer or FIFO (first-in, first-out) queue
People call me an animal
Since I possess a tail
but its in opposite direction.
Not sure about the animal part. The tail can end up in front of the head once the head wraps around after having reached the end of the buffer.
I cant travel beyond some point
Since i am bound by my tail.
A ring buffer is considered full when head = tail-1 (mod N). At this point, the head can no longer advance and so you can no longer push data to the ring buffer.
Sometimes I don't have a tail
But still I can not travel that far.
The tail points to stale data once it has caught up to the head, i.e. head = tail. Even starting with no tail, the head can only advance by N, where N is the (finite) size of the ring buffer.
Go to right and world is easy for you.
Go to left and change your mode.
A ring buffer always travels to the right, towards increasing memory, until wrap-around occurs after a total of N steps.
You touch me often, but I won't bite you.
A ring buffer can be used many times per second, allowing information to transfer at rates of 100s of MB per second.