Vågmästare is, like many of the best Swedish words, a compound noun.
Våg has two meanings: 'wave', as in the movement of the ocean but also used figuratively, and 'scales', as in the device used to weigh things. It's the second meaning we're interested in here.
Mästare means 'master', and shares its origins with the English word.
So en vågmästare means 'the master of the scales' and in the 18th and 19th centuries, it was used very literally. Sweden at the time had many weighing houses (järnvåg) where iron (järn) was weighed and stored.
These days, neither the weighing houses nor the official role of vågmästare exist any longer. But you'll still hear the word vågmästare used, often in a political context.
Nowadays, it refers to those politicians or parties who have the power to decide the outcome of an election or vote, metaphorically tipping the scales to help one side reach a majority. It's a word that has been used in Swedish since the 1920s and can crop up a lot, since there are a lot of political parties in Sweden's parliament and a political system that favours alliances and minority governments.
When two large blocs fail to reach a majority (like in the last two elections), it's often down to a smaller party or bloc to decide which group to align with. In English, the term usually used is 'kingmaker'.
On Friday, four of Sweden's parties (the centre-left Social Democrats and Green Party, plus the centre-liberal Centre and Liberal parties) confirmed a government agreement, which looked like a step towards ending four months of political deadlock. But even with those four former rivals in agreement, the parties' MPs didn't have the majority needed in order for the deal to pass a parliamentary vote.
The right-wing Moderate Party and Christian Democrats were highly unlikely to support the deal, as were the far-right Sweden Democrats. The most likely way the agreement could get past parliament was for the Left Party's 28 MPs to agree not to vote it down. In other, more poetic words: the Left Party was the vågmästare. It was up to them to decide whether to tip the scales (spoiler alert: on Wednesday, the party's leader confirmed they would allow the Social Democrat-led government to go ahead).
Otherwise, in recent years it's often been the far-right Sweden Democrats who have been described as vågmästare. Their emergence as the third largest bloc has left neither of the two traditional sides able to form a majority, raising the question of whether one of the main blocs would need to rely on Sweden Democrat support to govern.
Of course, it's worrying for the larger blocs if a party that only a small percentage of voters support can end up having the final say in who gets to govern. Because of this, vågmästare tends to be used in a pejorative sense, usually when the opposition party is highlighting the influence this small party will get. But it can also be used in a mere descriptive sense by political observers.
An alternative phrase with the same meaning is tungan på vågen, literally meaning 'the tongue on the scales'.
Vänsterpartiet kan blir vågmästare i svensk politik, i stället för Sverigedemokraterna
The Left Party can become the kingmakers in Swedish politics, instead of the Sweden Democrats
Vi ser det som en fördel att vi har blivit vågmästare
We consider it an advantage that we have become the kingmakers