All answers are partially wrong - Maybe due the new ES6/ES7 specs - , read why:

First of all, in JavaScript, the representation of the number is 2^53 - 1 that is true for @Luke answer, we can prove that by running Number.MAX_SAFE_INTEGER that will show a big number, then we do log2 to confirm that the number of bits is the same :

Number.MAX_SAFE_INTEGER
// 9007199254740991

Math.log2(9007199254740991)
// 53

However, Bitwise operation are calculated on 32 bits ( 4 bytes ), meaning if you exceed 32bits shifts you will start loosing bits.

Welcome to Javascript!

The MDN reference is pretty good, there's nothing in these answers that isn't in that article. I suggest you read ECMA-262 4.3.19 to 21 to understand the difference between a number value, a number Type and a number Object (and probably the whole section to understand Types in general).

An important aspect of javascript is that it is not strongly typed. There is no floating point precision type, nor is there an integer type. There are values, Types and Objects.

Variables don't have a type, their values do. You can assign a value to a variable, and that value can have a Type of String, Number, Boolean, etc. Don't get confused between Types and Objects - you can't specify that a variable has a particular Type, you can only assign it a value that has a Type. e.g.

Copyvar num = '5'; // the value of num is a String Type
num = +num;    // its value is now a Number Type
num = num/2;   // its value is still a Number Type
               // but if printed looks like a float (2.5)

There is a Number object too (though rarely used) and javascript handily converts number primitives to objects for the sake of evaluating expressions:

Copy(5).toString();  // 5 is converted to a Number object
                 // its toString method is called, returning '5'

In the above, the grouping operator is required as otherwise the 'dot' will be interpreted as a decimal place and the following 't' as a syntax error. It is essentially the reverse of parseInt.

The parseInt and parseFloat methods are used to convert a string Type value to a number Type value as shown in the MDN article. There are other methods, such as:

Copyvar num = '5';    // string Type
var x = num * 2;  // number Type

Variable x has a number Type value because multiplication will convert the string value of num to a number before doing multiplication.

Copyvar y = x + 'px';  // string Type

The '+' operator is overloaded - because one of the terms in the expression is a string, it will be treated as a concatenation operator so x will be converted to a string Type, 'px' will be concatenated and the string '10px' will be assigned to y, so its value is a string Type.

Read relevant parts of ECMA-262, it explains how it all works in detail.