A form is the only way, traditionally, to send data back to the server from the client. Any element inside the form which has a name attribute will be sent to the server when the user clicks the form's submit button, and the server can use the value of any of those elements. The programmer may hard-code the value attribute into the element and the user would not be allowed to change it, such as for checkboxes, radio buttons, and disabled text controls, or could allow the user to change the value, such as for regular text controls. If the programmer does not hard-code the value attribute and it is not an element that allows the user to change it, I believe it gets the values "true", if it is enabled, and "false", if it is disabled. "Enabled" and "disabled" may mean different things for different elements.
HTML before 5 required all of these elements to be in some form in order for the server to obtain their values, and it only got the values of the elements in the form the submit button was associated with, whether or not the elements had a name or value attribute. HTML5 still required the elements to be associated with a form to be submitted to the server, but they do not need to be inside the form anymore. HTML5 has ways for this to happen, usually by adding a form*something* attribute to the relevant elements.
My question is, why did all this come about? What is so special about forms that they became pretty much the only way to send data to a server until recently?
<form> defines an easy boundary for the user agent to be able to identify all the elements to be submitted to the server. It also allows the user agent to attach convenient default behaviors to the form and the form's child elements. For example being able to hit enter and the form data is submitted. It also allowed for a place to specify where the data would go via attributes on the form element. So all this behavior is available by default without JavaScript. At the same time it also allows easier access in JavaScript via the DOM (form.elements collection)
Why it has changed is because a significant number of sites are now using AJAX to submit the data, and the need for the default behavior is unnecessary in these cases. Often form may be included only as a formality and have no relevant attributes.
So in HTML5 they've allowed the old pattern as well as expanded the capability for developers who may be using AJAX and not need the default behaviors. Or for designers who may need flexibility in where they are placing their form elements (outside of the traditional hierarchy), while at the same time creating connections and keeping semantics alive.