How do I know and decide when I should be using a $.getJSON() or $.getJSONP() or $.ajax() method for requests?

$.getJSON() is shorthand usage of $.ajax(). For example:

$.getJSON()

is equal to

$.ajax({
  dataType: "json",
  url: url,
  data: data,
  success: success
});

Use getJSON if you need to fetch a json file from your site.

You are not allowed to just load json from other peoples sites for security reasons, so if you need to do that then you will probably need to specify that you want jsonp (JSON with Padding) and the site will have to return the json data in a wrapper. I'm not aware of a $.getJSONP function, but you can specify that you expect jsonp data returned using either $.getJSON() or $.ajax().

Use $.ajax() if you you need to specify some custom configuration that you don't get with the shorthand. The jQuery API will give you all of the details and options available http://api.jquery.com/jquery.ajax/

building on @LaVaughn Haynes answer there is not a $.getJSONP() function but in your $.ajax you will have a option to call a jsonp service.

$.ajax({
url: "http://notmydomain.com/api",
jsonp: "callback",
datatype: "jsonp",
data: {q: "select stuff", format: "json"},
success: function( data) {
    console.log(data) ; 
   }
}); 

you would jsonp mainly when you are doing across domain calls, but this is tricky as it's a main tactic for cross site scripting attacks.