In an inner join, the ordering of the tables in the join doesn't matter - the same rows will make up the result set regardless of the order they are in the join statement.
In either a left or right outer join, the order DOES matter. In A left join B, your result set will contain one row for every record in table A, irrespective of whether there is a matching row in table B. If there are non matching rows, this is likely to be a different result set to B left join A.
In a full outer join, the order again doesn't matter - rows will be produced for each row in each joined table no matter what their order.
Regarding A left join B vs B right join A - these will produce the same results. In simple cases with 2 tables, swapping the tables and changing the direction of the outer join will result in the same result set.
This will also apply to 3 or more tables if all of the outer joins are in the same direction - A left join B left join C will give the same set of results as C right join B right join A.
If you start mixing left and right joins, then you will need to start being more careful. There will almost always be a way to make an equivalent query with re-ordered tables, but at that point sub-queries or bracketing off expressions might be the best way to clarify what you are doing.
As another commenter states, using whatever makes your purpose most clear is usually the best option. The ordering of the tables in your query should make little or no difference performance wise, as the query optimiser should work this out (although the only way to be sure of this would be to check the execution plans for each option with your own queries and data).