Landing pages are custom designed pages that have major call to action to the user that lands on it. Call to action can be calling a phone number, filling out a contact form, or clicking to a product. The landing page's main purpose is to capture the audience and entice them to commit to the call to action. Landing pages can have multiple calls to action, but usually just one main strong one.
Your home page is usually your main landing page, but in most cases when done right you would be using both search engine marketing and organic hits to drive customers to the specific custom tailored page. Landing pages do not need to be over functional beyond the main call to action, this is why they are a great tool to not clutter up your standard home page.
Web pages are meant to be informational and easy to navigate. If your home page is cluttered and looks like a car salesman ad, it would be a good time to build a solid landing page to increase conversion and lower your main websites bounce rate.