Short URL
A short URL is an web page address (technically a URL) that can be expressed with minimal size. It is the shorter in a set of two or more synonyms, that is, a set of URLs pointing to the same page. For instance, the address
http://osm.org/relation/59470
is shorter tham
http://www.openstreetmap.org/relation/59470.
Importance and applications
Web browsers support URLs of any size, but humans need to audit URLs to make sure they are browsing where they believe they are safe. In general, it is enough to see the domain name, and in fact, the trust that the public places in a domain is the most valuable asset of a brand on the internet. In the case of OpenStretMap there are a trust over OPENSTREETMAP.ORG, and some growing perception of equivalence over OSM.ORG, so transferring some trust to it.
There are some contexts where the complete address is also important to be audited by humans, and, naturally, will be easy to audit a shorter URL. In the context of OSM maps, is important to ensure the URL of a map feature.
Shorter URLs will be easy to show in printed media, easy to write in an e-mail without a line-break... And will be the only valid URL in a media like Twitter or to express as viable QR code symbol.
Short URL redirection
When we type in the web browser the short address of the OpenStreetMap, http://OSM.ORG, the browser quickly transforms the URL in a new one, https://www.openstreetmap.org. This transformation, named "URL forwarding" or URL redirection, is a fundamental Web mechanism, technically knowed also as HTTP status 301. The URL that was quickly transformed into the final one, is said "an alias" of the final one.
The alias can be used also to express human-readable URL (shorter or not), specified by W3C as a Cool URI, that will be the canonical URL, to be used in advertising or "official URL publishing". For example: the canonical URL for "httpS://Schema.Org/Person" is http://schema.org/Person.
In general is used to hide non-permanent or ugly details of the common URL. For example this page can be accessed by an ugly URL as https://wiki.openstreetmap.org/w/index.php?title=Short_URL
So, a "short redirection" can be a simple alias for a "long URL", but in general is also a "cool URL", and, eventually, can be the canonical URL of a web page or web resource (like in a download page).
Short PURL
PURL is an abbreviation for "Persistent URL", a kind of URL that is "eternal"... There are some moral or contractual compromise of the domain name's owner that the URL will be "never" changed, never become unavailable or an "Error HTTP 404".
The most commom mechanism to implement a PURL, and used since 1995 by the PURL's original purpose, is to redirect (by HTTP status 301) the canonical URL to the current URL.
In a "web ecosystem" perspective, the PURL is a digital preservation mechanism that preserves the hyperlinks and hypertexts on the Web. Preserves and enhances the integrity of any web content.