We begin a new tractate, Sanhedrin, which discusses court cases. Unlike in many secular court systems, the judges are active participants (they're the ones who question witnesses) and the ultimate decisors; there are no lawyers or juries.
A court is made up of some number of judges, depending on the type of case (at least 3, sometimes 23 or 71 or occasionally other numbers). Here are some of the cases listed in the first mishna of the tractate (this is not a complete list):
Various types of monetary damages are judged by three.
Rape, seduction, and libel require three according to R' Meir, but the sages say libel requires 23 because it could involve a capital charge. (A note suggests this comes up with adultery but doesn't connect the dots. Also, rape and seduction can involve capital charges too, so I don't know why they only call out libel. Perhaps it's addressed later in the g'mara.)
Capital cases, as implied in the previous bullet, require 23.
Cases for which the punishment is flogging require three, but according to R' Yishmael, 23.
Calendar decisions (witnessing the new moon, adding a leap month) are judged by three, though R' Shimon b. Gamaliel describes a more complicated scheme.
A tribe charged with idolatry, a false prophet, and a high priest can be tried only by a court of 71.
The following require 71: authorizing wars of free choice, adding to the temple courtyards, establishing small sanhedrins (of 23) for the tribes, condemning a city, condemning frontier towns.
Why is a great sanhedrin 71? Because Moshe was commanded to gather 70 (other) men. And why is a small sanhedrin 23? It's complicated. (I don't completely follow their math, sorry.)
This is all from 2a. The mishna continues onto 2b before the g'mara starts there.
(Today's daf is 4.)