a) Neighborhood - Essentially, the school you can walk to. It is always open to you and you are guaranteed a spot there. It might have specialized programs (i.e. Kensington Health Sciences) or it might not (i.e. West Philadelphia High School).
b) Magnet - A school with criteria for entry. You must pass a test, audition, have high test scores, or other admissions requirements. There are a variety of specialized programs (i.e. the Creative and Performing Arts High School) or some that have just been generally "good" for years (i.e. Central High School).
c) City-wide lottery - Similar to a magnet school that you have to apply, but there are few - if any - criteria for entry. Names of applicants are pulled out of a lottery (I think the High School of the Future still works this way).
In contract, New York Department of Education schools work differently. From my understanding, every high school is essentially a "neighborhood" and "magnet" school simultaneously. Students are guaranteed a spot at their local, so-called "zoned" school (that is, unless Mayor Bloomberg gets his way)., but can apply to any high school in the city. This confuses me quite a bit as I am not used to it at all. But it is a fascinating different policy decision that they made years ago.
Here is to more comparisons in the future!