The mishna begins chapter 4 of Sanhedrin with an overview of how civil and capital cases are conducted:
Both civil and capital cases require inquiry and examination of witnesses. (This is done by the judges; there are no lawyers.)
Civil cases are tried by a court of three; capital cases are tried by a court of 23.
When the judges deliberate on civil cases, they may begin with arguments for either acquittal or condemnation. When they deliberate on capital cases, they must begin with arguments for acquittal.
Civil cases may be decided by a majority of one; capital cases may be decided by a majority of one for acquittal, but require a majority of at least two for condemnation.
In civil cases the decision may be reversed in either direction (for example upon the discovery of an error). In capital cases the decision may be reversed from condemnation to acquittal but not the other way around.
In civil cases, all present (including the pupils who are observing) may argue for or against the defendant. In capital cases, anybody may argue for acquittal but only the judges may argue for condemnation.
In civil cases, one who has previously argued for either acquittal or condemnation may then argue for the other side (for example because he realized his argument was faulty). In capital cases, one who has argued for condemnation may then argue for acquittal but not the other way around.
Civil cases are tried by day and concluded by night if necessary. Capital cases are tried by day and must be concluded by day. Civil cases can be concluded on the same day (either way); capital cases can be concluded on the same day for acquittal but not until the following day for condemnation. Therefore trials are not held on the eve of Shabbat or a festival.
In civil cases we begin with the opinion of the most eminent of the judges; in capital cases we begin with the opinion of the least ("those on the side benches").
All types of Jews (presumably they mean men) are eligible to try civil cases, but converts and bastards cannot judge capital cases.