Number refers to a global constructor function that, when called, returns a number (or, in very strange cases, a Number object).

number is the string returned by the typeof operation - or, the word one uses to describe a number primitive.

It's only a capitalization difference, but it's quite an important distinction.

const str = '34';
const num = Number(str);
console.log(typeof num);

What exactly is this Number object? Is it an instance of some class?

Number itself is a class, kind of - a function that can be called, possibly with new, which will return a value that inherits from Number.prototype. (Although it's not syntactically forbidden, best to never use new with Number)

Unlike some languages, JavaScript types aren't really values you can work with and manipulate inside a script. You can say, when looking at a piece of code: "This variable points to a number" - and you can use the typeof operator to extract the string representation of the type from a value - but that's it. In actual JavaScript code, there isn't a number value or type that can be referenced in the code to do something with (other than when as part of a string in conjunction with the use of typeof). This is in contrast with, for example, TypeScript, where you could do type NumObj = { theValue: number } and declare const theMap: Map<number, string>, and lots of other interesting things - but JavaScript only has the 'number' that typeof returns, and the Number constructor and its methods.

Number is a constructor function, but it also has static methods - methods directly on the constructor, not on the prototype object, such as isInteger - a utility function related to numbers that isn't that particular to a given number instance.

UPDATE: with a new ES2020 standard released this answer is not entirely correct anymore, see the other answer (from @Mathias Lykkegaard Lorenzen) about BigInt details.

There is only the Number data type in JS that represents numbers.

Internally it is implemented as IEEE 754 double precision floating point number.

What it means is that - technically there is no dedicated data type that represents integer numbers.

Practically it means that we can safely use only numbers that are safely representable by the aforementioned standard. And it includes integer values in the range: [-9007199254740991; 9007199254740991]. Both values are defined as constants: Number.MIN_SAFE_INTEGER and Number.MAX_SAFE_INTEGER correspondingly.