Elections where the winning party lost the popular vote.

[Famous Mortimer]

The HUGE one is South Africa 1948, when the National Party won despite losing the popular vote by 10%.

[Famous Mortimer]

Also, in Sri Lanka 1970, the United National Party won a plurality but actually came in third in terms of seats.

This was due to an electoral pact between the Sri Lanka Freedom and a Trotskyite party. That alliance together won even more than UNP, so it wasn't totally unfair in this case.

[Famous Mortimer]

I know a few election cycles ago in Israel, Kadima won the most seats by a narrow margin over Likud, but the right wing had scored a clear majority in the Knesset. Thus despite having a plurality, Livni wasn't asked to form a government as she had no chance at success.

This seems like it's more common in a parliamentary system, where it's the winning bloc that matters, not the party.

That's a different issue entirely and it's way more common. Social Democrats in Scandinavia basically always come in first in every election but they don't always form a government for this reason.

I think OP was talking about winning the most seats (regardless of whether they form a government or not) but getting less votes than another party. In order for this to happen, you basically HAVE to have FPTP, or at least mixed member PR with some of the members elected via FPTP.

The recent Ukrainian election saw the Popular Front win the most votes of any party but the Poroshenko Bloc won more seats because they were a close second but won way more FPTP constituency seats. Even this is somewhat less dramatic and outrageous though as when you combine the party list votes with the constituency votes, Poroshenko Bloc deserved to come in first.

Really truly undemocratic sh!t only happens with pure FPTP.

[Famous Mortimer]

There are generally two categories of circumstances where this happens.

One is where the districts are malapportioned so that there are more voters per seat in the jurisdictions favoring the popular vote winner. It can be due to constitutional apportionments, so it becomes a structural feature of that legislature. It can also be due to party strategies in multimember districts. This tends to be the cause of a spurious majority in PR-based elections. Its occurrence in US districts before 1960 gave rise to the one-man-one-vote decisions.

The other one is where the partisan voters of one party are inefficiently distributed. It can be a natural geographic occurrence, such as when one party has a primarily urban (or rural) base. It can also be the result of successful gerrymandering to pack one party's base into as few districts as possible. This tends to be the cause in FPTP-based elections. OH in 2012 is a good example of both gerrymandering and natural inefficiency combining to give the Pubs 12 of 16 House seats as Obama won the state overall.

Not running a full slate of candidates can help as well.

Yep, that's what happened in the Sri Lanka case I mentioned.

[Famous Mortimer]

The Georgia Gubernatorial Election of 1966

Lester Maddox (Dem) - 450,626
Bo Callaway (Rep) - 453, 665
Ellis Arnell (Dem, write-in) - 69,025

A bitter Democratic primary between Maddox and Arnell - representing the conservative and liberal wings of the party, respectively - led to Arnell leading an insurgent write-in campaign.  At the same time, Callaway, marking a resurgent Republican Party in Georgia, became the party's first viable candidate for Governor in living memory.

As a result, the three-way contest failed to produce a winner with an overall majority.  At the time, if a candidate failed to win a majority at the election, the General Assembly will choose the governor - much as the federal House of Representatives still does under the electoral college. 

Of course, Democrats made up the overwhelming faction of the legislature at the time and easily chose Maddox as Governor, even though he won 3,000 fewer votes than Callaway.

EDIT: Just remembered that this is technically the International General Discussion board, but I thought this result was too interesting to fail to mention.

Bolivia had a similar system where the Congress chose between the top 2 if no one got a majority. I don't know how many times (if ever) they picked the second place winner.

[Famous Mortimer]

Yeah, now I'm remembering. Bolivia '89 was particularly famous and horrible. A non sensical congressional coalition of left and right was formed to oust the centrist candidate.

[Famous Mortimer]

Southern Rhodesia 1933:

https://en.wikipedia.org/wiki/Southern_Rhodesian_general_election,_1933