When liquids simmer on a stovetop, they bubble quietly without actually getting hot enough to boil.
So, figuratively, when people simmer, they keep on feeling bad emotions (like anger or frustration) for a long time, at a low or medium intensity, often in a hidden way.
And when bad things and bad feelings simmer, they keep on existing for a long time, at a low or medium intensity, often in a hidden way.
Lastly, "simmer down" is a slangy way to say "calm down."
Part of speech:
Usually a verb, the intransitive kind: "he simmers in anger," "her frustration simmers."
But it can also be a noun: "he's in a simmer," "the anger is still in a simmer."
Other forms: simmered, simmering
How to use it:
"Simmer" usually has a negative tone. Use it to describe things, feelings, and people that seem like they might boil over or explode soon, yet they keep on stewing in a slow, tense, passive way.
Say that a person, a group, a place, or a bad feeling (like anger or frustration) is simmering: "the team simmered over their defeat," "resentment simmers in their neighborhood."
Or, a person can simmer with or in some emotion: "she's simmering with frustration," "he simmered all weekend in jealous anger."
Because simmering is less intense than boiling--or maybe because you often keep a lid on your food while you're letting it simmer--we can say that something simmers under or beneath something else.
And when you need an adjective, use "simmering;" talk about simmering insults or affronts, simmering tensions and arguments, simmering conflicts and wars, simmering anger and frustration, simmering questions and unknowns, etc.
Through all those years as caretakers, unpaid, unthanked, and unwilling, they simmered with resentment.
"[The television series The Americans] has always simmered and stewed in the emotional strain of its premise without ever actually exploding."
—Lili Loofbourow, Slate, 31 May 2018