What are the Rust types denoted with a single apostrophe?

These are Rust's named lifetimes.

Quoting from The Rust Programming Language:

Every reference in Rust has a lifetime, which is the scope for which that reference is valid. Most of the time lifetimes are implicit and inferred, just like most of the time types are inferred. Similarly to when we have to annotate types because multiple types are possible, there are cases where the lifetimes of references could be related in a few different ways, so Rust needs us to annotate the relationships using generic lifetime parameters so that it can make sure the actual references used at runtime will definitely be valid.

Lifetime annotations don’t change how long any of the references involved live. In the same way that functions can accept any type when the signature specifies a generic type parameter, functions can accept references with any lifetime when the signature specifies a generic lifetime parameter. What lifetime annotations do is relate the lifetimes of multiple references to each other.

Lifetime annotations have a slightly unusual syntax: the names of lifetime parameters must start with an apostrophe '. The names of lifetime parameters are usually all lowercase, and like generic types, their names are usually very short. 'a is the name most people use as a default. Lifetime parameter annotations go after the & of a reference, and a space separates the lifetime annotation from the reference’s type.

Said another way, a lifetime approximates the span of execution during which the data a reference points to is valid. The Rust compiler will conservatively infer the shortest lifetime possible to be safe. If you want to tell the compiler that a reference lives longer than the shortest estimate, you can name it, saying that the output reference, for example, has the same lifetime as a given input reference.

The 'static lifetime is a special lifetime, the longest lived of all lifetimes - for the duration of the program. A typical example are string "literals" that will always be available during the lifetime of the program/module.

You can get more information from this slide deck, starting around slide 29.

Lifetimes in Rust also discusses lifetimes in some depth.

To add to quux00's excellent answer, named lifetimes are also used to indicate the origin of a returned borrowed variable to the rust compiler.

This function

pub fn f(a: &str, b: &str) -> &str {
  b
}

won't compile because it returns a borrowed value but does not specify whether it borrowed it from a or b.

To fix that, you'd declare a named lifetime and use the same lifetime for b and the return type:

pub fn f<'r>(a: &str, b: &'r str) -> &'r str {
//      ----              ---         ---
  b
}

and use it as expected

f("a", "b")