What makes something a "trait object"?

Recent Rust changes have made "trait objects" more prominent to me, but I only have a nebulous grasp of what actually makes something into a trait object. One change in particular is the upcoming change to allow trait objects to forward trait implementations to the inner type.

Given a trait Foo, I'm pretty sure that Box<Foo> / Box<dyn Foo> is a trait object. Is &Foo / &dyn Foo also a trait object? What about other smart-pointer things like Rc or Arc? How could I make my own type that would count as a trait object?

The reference only mentions trait objects once, but nothing like a definition.

You have trait objects when you have a pointer to a trait. Box, Arc, Rc and the reference & are all, at their core, pointers. In terms of defining a "trait object" they work in the same way.

"Trait objects" are Rust's take on dynamic dispatch. Here's an example that I hope helps show what trait objects are:

// define an example struct, make it printable
#[derive(Debug)]
struct Foo;

// an example trait
trait Bar {
    fn baz(&self);
}

// implement the trait for Foo
impl Bar for Foo {
    fn baz(&self) {
        println!("{:?}", self)
    }
}

// This is a generic function that takes any T that implements trait Bar.
// It must resolve to a specific concrete T at compile time.
// The compiler creates a different version of this function
// for each concrete type used to call it so &T here is NOT
// a trait object (as T will represent a known, sized type
// after compilation)
fn static_dispatch<T>(t: &T)
where
    T: Bar,
{
    t.baz(); // we can do this because t implements Bar
}

// This function takes a pointer to a something that implements trait Bar
// (it'll know what it is only at runtime). &dyn Bar is a trait object.
// There's only one version of this function at runtime, so this
// reduces the size of the compiled program if the function
// is called with several different types vs using static_dispatch.
// However performance is slightly lower, as the &dyn Bar that
// dynamic_dispatch receives is a pointer to the object +
// a vtable with all the Bar methods that the object implements.
// Calling baz() on t means having to look it up in this vtable.
fn dynamic_dispatch(t: &dyn Bar) {
    // ----------------^
    // this is the trait object! It would also work with Box<dyn Bar> or
    // Rc<dyn Bar> or Arc<dyn Bar>
    //
    t.baz(); // we can do this because t implements Bar
}

fn main() {
    let foo = Foo;
    static_dispatch(&foo);
    dynamic_dispatch(&foo);
}

For further reference, there is a good Trait Objects chapter of the Rust book