A bio-diesel refinery is really not a refinery, but basically a bucket processor that uses great big huge buckets. Refiners also typically get their stock as freshly produced vegetable oils. Smaller "bucket" processors may obtain their raw materials from various sources.

A bio-diesel manufacturer may also deal in such a huge volume that they can use a continuous process.

Now, there is also something known a vegetable oil refining. This is the same as oil refining, except that they use vegetable oil along with crude oil. The results are the same petrol-based products we are already familiar with (eg. gasoline, propane, plastics).

I compared a bio-diesel "refinery" to a bucket processor here because they are the most similar, but maybe you are talking about an actual petrol-chemicals refinery?

Refineries are basically great big chemistry labs which perform all sorts of chemical reactions, but on a massive scale. Crude oil is a mixture of many different organic compounds. Mostly long-chain hydrocarbons between 6 and 20 carbon atoms in length. One of the first steps at a refinery is to sort the oil into similar lengths by using what is basically a still. Once the oil is separated into batches of uniform molecular weight, it can undergo further processing. For example, long chains (12+ carbon atoms) might be sent through a hydrogen cracker, which turns the long chains into shorter ones.

The fatty acids that make up plant oils are also hydrocarbon chains of 3 to 21 carbon atoms long; however, they come with extra bits on one end. Bio-diesel processors basically remove these bits, and then cap the chains. They do no sorting, cracking, or other alterations.... although they could. In fact, if bio-diesel was distilled, you could make bio-gasoline out of the shorter chains, and keep the longer chains for bio-diesel and bio-kerosene. This is basically what is done in the afore mentioned vegetable oil refining.