We consider a queue with finite buffer where the buffer size limits the amount of work that can be stored in the queue. The arrival process is a Poisson or a Markov modulated Poisson process. The service times (packet lengths) are i.i.d. with a general distribution. Our queue models the systems in the Internet more realistically than the usual M/GI/1/K queue which restricts the number of packets in the buffer rather than the buffer content (the number of bits). We obtain the stability, the rates of convergence to the stationary distribution and functional limit theorems for this system. In addition, we also obtain algorithms to compute the stationary density of the workload process, the waiting times and the probability of packet loss. Next, we study the queue with two priority classes. The higher priority traffic has preemptive-resume priority. For sharing the buffer, we consider two cases. In the first case, the buffer is shared by both the classes without any priority. In the second case, the buffer is partitioned into two groups, one reserved for each class. For this system also we obtain all the results mentioned for the single class traffic.