No. I simply don't know what "unelected" means in the context of British PMs, nor why it's important.
Here's how the process of election works for the UK (and most other Commonwealth countries). Forgive me if I am going over ground you know.
1) Members of Parliament (MPs) are elected by the people in much the same way as congresspeople in the USA. They win a "seat" in parliament in an election.
2) In the lower house of parliament, called Congress in the USA and The Members in UK parliament, the party that commands the majority of members gets to be "the Queen's government". This majority may require forming a partnership with another smaller party to get the numbers. The non-government members become The Loyal Opposition.
3) On the government side, various members are chosen to manage various businesses of government - Treasury, state affairs, foreign affairs, etc. They are called Ministers (of the Crown - QE2 will take their oath of office) and they are equivalent to US Secretaries. The method of choosing is up to the party in government, not directly by the people. It may be a internal party election, favours for the boys, or even WWE Royal Rumble style. Whatever.
4) On the government side, one of the members is chosen to be their notional leader - the Prime Minister (PM). In some Commonwealth countries this is called First Minister. Again, the method of choosing a PM is up to the party in government, not the people. Thus the PM is not directly elected by the people. The PM is charged with leading the government side and thus the parliament, and therefore the country. The PM usually gets to choose Ministries and Ministers to run them.
5) During a government's term, the terms of all ministers including the PM are subject to the whims of the governing party (or alliance). The PM and indeed any Minister may be tossed out of the position by party politics or other factors. They will be replaced by someone the party now likes more. In that sense, the replacement PM is still not elected by the people. They get to like or lump what the government chooses.
This PM replacement process has happened a few times recently. In Australia, we are making it a regular thing for both sides of politics. Mrs May in the UK has just resigned as PM (because her government has become so rambunctious about Brexit that they have no idea what they are doing...but they don't want her). She will revert to being just an MP, and some other MP will be chosen to replace her. This all without reference to the people directly.
That's a vastly simplified description, because like cricket there are many more subtleties involved. But hopefully that helps.