gemäßigt
temperate
Describes a climate that is neither extremely hot nor extremely cold, but has moderate temperatures.
Deutschland liegt in der gemäßigten Klimazone.
Germany lies in the temperate climate zone.

moderate
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temperate
Describes a climate that is neither extremely hot nor extremely cold, but has moderate temperatures.
Deutschland liegt in der gemäßigten Klimazone.
Germany lies in the temperate climate zone.

The word 'gemäßigt' (moderate) is the past participle of the verb 'mäßigen' (to moderate/to restrain) and belongs to the word family of 'das Maß' (measure). It literally means that something has a good measure and is not extreme.
Er trinkt Alkohol nur in gemäßigten Mengen.
He drinks alcohol only in moderate amounts.
In connection with weather, 'gemäßigt' (temperate) is almost always used in combination with 'Klima' (climate), 'Klimazone' (climate zone), or 'Temperaturen' (temperatures).
Die meisten Menschen bevorzugen gemäßigte Temperaturen im Sommer.
Most people prefer moderate temperatures in summer.
When it comes to opinions, 'gemäßigt' (moderate) describes a non-extreme attitude. Typical combinations are 'eine gemäßigte Position' (a moderate position), 'eine gemäßigte Meinung' (a moderate opinion), or 'ein gemäßigter Politiker' (a moderate politician).
In der Diskussion vertrat sie eine sehr gemäßigte Haltung.
In the discussion, she took a very moderate stance.
In reference to behavior or actions, 'gemäßigt' (moderate) is often used as a polite request or description meaning 'not too fast' or 'not too much.' Common combinations are 'gemäßigtes Tempo' (moderate speed) and 'gemäßigter Preis' (moderate price).
Ein gemäßigtes Training ist gut für die Gesundheit.
Moderate training is good for your health.
middle / center
'die Mitte' and 'gemäßigt' can both mean in politics or opinions that something is not extreme. 'die Mitte' is a noun and means a political direction between left and right. 'gemäßigt' is an adjective and describes an attitude, opinion, or person that is not radical. So you would say, for example, 'eine Partei der Mitte' (a centrist party), but 'eine gemäßigte Meinung' (a moderate opinion). 'Mitte' tends to name the position, while 'gemäßigt' tends to describe the nature of the attitude.

Der Politiker bleibt in der Diskussion ruhig und vertritt eine gemäßigte Meinung.
The politician remains calm in the discussion and represents a moderate opinion.
Die Partei sieht sich politisch in der Mitte.
The party sees itself as being in the political center.