Bach wrote his six Brandenburg Concertos for the Margrave of Brandenburg, Christian Ludwig, possibly hoping to secure employment in the margrave's court. Lacking the necessary virtuosic players, the margrave never heard the concertos, and the manuscripts weren't published until 1850, one-hundred years after Bach's death.
Typically, when we think of a "concerto" we think of an orchestra along with a soloist. The Brandenburgs are so-named for the woven web of solos passed around the ensemble throughout each piece.
If a composer, like Bach, chooses to use multiple soloists from inside the orchestra, rather than one soloist in front of the ensemble, it is usually referred to as a concerto grosso. Bach, likely aware he was bending the traditional concerto grosso rules in half of his Brandenburgs, instead called them Concerts avec plusieurs instruments, or Concertos with several instruments.
Strict rules aside, it is acceptable to refer to these as concerti grossi. The soloists in a concerto grosso are referred to as the concertino, which alternates with the ripieno, or remainder of the ensemble.
Each concerto stands apart from the other, using quite the variety of instruments available at the time. As a result, we often colloquially refer to Brandenburgs by their instruments (i.e. the trumpets one, the violas one, the recorders one, etc.) Often, if unsure which Brandenburg is playing, you can eliminate possibilities based on instrumentation.
Thank the Internet for providing the ability to view all of Bach's scores to these works.