How to know that a string starts/ends with a specific string in jQuery?

You do not really need jQuery for such tasks. In the ES6 specification they already have out of the box methods startsWith and endsWith.

var str = "To be, or not to be, that is the question.";
alert(str.startsWith("To be"));         // true
alert(str.startsWith("not to be"));     // false
alert(str.startsWith("not to be", 10)); // true

var str = "To be, or not to be, that is the question.";
alert( str.endsWith("question.") );  // true
alert( str.endsWith("to be") );      // false
alert( str.endsWith("to be", 19) );  // true

Currently available in FF and Chrome. For old browsers you can use their polyfills or substr

One option is to use regular expressions:

if (str.match("^Hello")) {
   // do this if begins with Hello
}

if (str.match("World$")) {
   // do this if ends in world
}

For startswith, you can use indexOf:

if(str.indexOf('Hello') == 0) {

...

ref

and you can do the maths based on string length to determine 'endswith'.

if(str.lastIndexOf('Hello') == str.length - 'Hello'.length) {

There is no need of jQuery to do that. You could code a jQuery wrapper but it would be useless so you should better use

var str = "Hello World";

window.alert("Starts with Hello ? " + /^Hello/i.test(str));        

window.alert("Ends with Hello ? " + /Hello$/i.test(str));

as the match() method is deprecated.

PS : the "i" flag in RegExp is optional and stands for case insensitive (so it will also return true for "hello", "hEllo", etc.).