What is the difference between a URI, a URL and a URN?

URIs identify and URLs locate; however, locators are also identifiers, so every URL is also a URI, but there are URIs which are not URLs.

Examples

• Roger Pate

This is my name, which is an identifier. It is like a URI, but cannot be a URL, as it tells you nothing about my location or how to contact me. In this case it also happens to identify at least 5 other people in the USA alone.

• 4914 West Bay Street, Nassau, Bahamas

This is a locator, which is an identifier for that physical location. It is like both a URL and URI (since all URLs are URIs), and also identifies me indirectly as "resident of..". In this case it uniquely identifies me, but that would change if I get a roommate.

I say "like" because these examples do not follow the required syntax.

Popular confusion

From Wikipedia:

In computing, a Uniform Resource Locator (URL) is a subset of the Uniform Resource Identifier (URI) that specifies where an identified resource is available and the mechanism for retrieving it. In popular usage and in many technical documents and verbal discussions it is often incorrectly used as a synonym for URI, ... [emphasis mine]

Because of this common confusion, many products and documentation incorrectly use one term instead of the other, assign their own distinction, or use them synonymously.

URNs

My name, Roger Pate, could be like a URN (Uniform Resource Name), except those are much more regulated and intended to be unique across both space and time.

Because I currently share this name with other people, it's not globally unique and would not be appropriate as a URN. However, even if no other family used this name, I'm named after my paternal grandfather, so it still wouldn't be unique across time. And even if that wasn't the case, the possibility of naming my descendants after me make this unsuitable as a URN.

URNs are different from URLs in this rigid uniqueness constraint, even though they both share the syntax of URIs.

From RFC 3986:

A URI can be further classified as a locator, a name, or both. The term "Uniform Resource Locator" (URL) refers to the subset of URIs that, in addition to identifying a resource, provide a means of locating the resource by describing its primary access mechanism (e.g., its network "location"). The term "Uniform Resource Name" (URN) has been used historically to refer to both URIs under the "urn" scheme [RFC2141], which are required to remain globally unique and persistent even when the resource ceases to exist or becomes unavailable, and to any other URI with the properties of a name.

So all URLs are URIs, and all URNs are URIs - but URNs and URLs are different, so you can't say that all URIs are URLs.

If you haven't already read Roger Pate's answer, I'd advise doing so as well.