What is the HTML for="" attribute in <label>?

I feel the need to answer this. I had the same confusion.

<p>Click on one of the text labels to toggle the related control:</p>

<form action="/action_page.php">
  <label for="female">Male</label>
  <input type="radio" name="gender" id="male" value="male"><br>
  <label for="female">Female</label>
  <input type="radio" name="gender" id="female" value="female"><br>
  <label for="other">Other</label>
  <input type="radio" name="gender" id="other" value="other"><br><br>
  <input type="submit" value="Submit">
</form>

I changed the for attribute on the 'male' label to female. Now, if you click 'male' the 'female' radio will get checked.

Simple as that.

To associate the <label> with an <input> element, you need to give the <input> an id attribute. The <label> then needs a for attribute whose value is the same as the input's id:

<label for="username">Click me</label>
<input type="text" id="username">

The for attribute associates a <label> with an <input> element; which offers some major advantages:
1. The label text is not only visually associated with its corresponding text input; it is programmatically associated with it too. This means that, for example, a screen reader will read out the label when the user is focused on the form input, making it easier for an assistive technology user to understand what data should be entered.
2. You can click the associated label to focus/activate the input, as well as the input itself. This increased hit area provides an advantage to anyone trying to activate the input, including those using a touch-screen device.

Alternatively, you can nest the <input> directly inside the <label>, in which case the for and id attributes are not needed because the association is implicit:

<label>Click me <input type="text"></label>

Notes:
One input can be associated with multiple labels.
When a <label> is clicked or tapped and it is associated with a form control, the resulting click event is also raised for the associated control.

Accessibility concerns

Don't place interactive elements such as anchors or buttons inside a label. Doing so, makes it difficult for people to activate the form input associated with the label.

Headings

Placing heading elements within a <label> interferes with many kinds of assistive technology, because headings are commonly used as a navigation aid. If the label's text needs to be adjusted visually, use CSS classes applied to the <label> element instead.
If a form, or a section of a form needs a title, use the <legend> element placed within a <fieldset>.

Buttons

An <input> element with a type="button" declaration and a valid value attribute does not need a label associated with it. Doing so may actually interfere with how assistive technology parses the button input. The same applies for the <button> element.

Ref: https://developer.mozilla.org/en-US/docs/Web/HTML/Element/label

The for attribute is used in labels. It refers to the id of the element this label is associated with.

For example:

<label for="username">Username</label>
<input type="text" id="username" name="username" />

Now when the user clicks with the mouse on the username text the browser will automatically put the focus in the corresponding input field. This also works with other input elements such as <textbox> and <select>.

Quote from the specification:

This attribute explicitly associates the label being defined with another control. When present, the value of this attribute must be the same as the value of the id attribute of some other control in the same document. When absent, the label being defined is associated with the element's contents.

As far as why your question is tagged with jQuery and where did you see it being used in jQuery I cannot answer because you didn't provide much information.

Maybe it was used in a jQuery selector to find the corresponding input element given a label instance:

var label = $('label');
label.each(function() {
    // get the corresponding input element of the label:
    var input = $('#' + $(this).attr('for'));
});