der Brei
porridge / purée
A soft, thick food made from cooked and often mashed grain, fruit, or vegetables.
Das Baby bekommt mittags einen Gemüsebrei.
The baby gets a vegetable purée at lunchtime.

porridge / mush
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porridge / purée
A soft, thick food made from cooked and often mashed grain, fruit, or vegetables.
Das Baby bekommt mittags einen Gemüsebrei.
The baby gets a vegetable purée at lunchtime.

The word 'Brei' (porridge/mush) is almost always used in the singular because it describes an uncountable mass. The plural form 'Breie' is very rare and is only used when referring to different kinds.
Zum Frühstück esse ich gerne einen warmen Brei.
For breakfast, I like to eat a warm porridge.
The idiom 'um den heißen Brei herumreden' (to beat around the bush) is a fixed structure. It means that someone avoids the real topic and does not get straight to the point.
Bitte rede nicht um den heißen Brei herum und sag mir die Wahrheit.
Please don't beat around the bush and tell me the truth.
In connection with food, 'Brei' (porridge/purée) is often used as the second part of a compound word that names the ingredient. Typical examples are 'Haferbrei' (oatmeal/porridge), 'Grießbrei' (semolina pudding), or 'Kartoffelbrei' (mashed potatoes).
Ein warmer Haferbrei mit Früchten ist ein gesundes Frühstück.
A warm oatmeal with fruit is a healthy breakfast.
When 'Brei' (mush/sludge) describes a shapeless mass, it often appears in connection with words like 'Matsch' (mud), 'Schnee' (snow), or 'Erde' (soil/dirt). It then describes an unpleasant, thick consistency.
Nach dem Tauwetter war der Weg ein Brei aus Schnee und Schmutz.
After the thaw, the path was a mush of snow and dirt.
mass / crowd
'die Masse' and 'der Brei' can both describe something soft, shapeless, and mushy. 'Masse' is more general and more neutral. It is used for many substances, for example dough, glue, or concrete, when the shape is not important. 'Brei' sounds more strongly like something thick or mushy and is often used for food or for something that has been crushed until very soft. Learners may confuse the words because both lack a fixed shape. As a practical rule: for a substance, people often say 'Masse'; for mushy food or a very soft mixture, 'Brei' is more likely.

Nach dem Kochen zerdrückt Lisa die Kartoffeln zu einem weichen Brei.
After cooking, Lisa mashes the potatoes into a soft mush.
Die Masse aus Mehl und Wasser muss noch zehn Minuten ruhen.
The mass made of flour and water still has to rest for ten minutes.