What is the difference between a binary star system and our star system?
A binary star system has two stars in the same system; our solar system has only one star.
Is the main difference between a pulsar and a binary system that a pulsar is made up of two neutron stars and that a binary system is made of of two normal stars?
Not exactly, while a pulsar is a specific type of neutron star (that being a "spinning neutron star") and a binary system is a pair of stars orbiting each other, a pulsar does not need to be part of a binary system. It would be possible, however to have a binary system with one of the pair being a pulsar and the other a neutron star (assumed to be the non-spinning or "normal neutron star"…
A solar system with two suns is referred to as a binary star system. Likewise, a system with three suns (very rare, as far as we know) is called a trinary star system. Systems orbiting binary stars are often just small asteroid clusters, heavily baked rocky planets, or odd gas giants, since the dynamics in a binary system are less stable than in a single-star system.