What is the soft part of the palm called in English?
I don't know the name even in my native language, so I cannot look it up in the dictionary.
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The anatomical term for it is the thenar eminence:
However, this is a scientific term, known mostly to medical students and doctors. It's not in general use. I had never even heard of it until I started googling just now. A few sources say thenar prominence. If you're wondering, both "eminence" and "prominence" come from a Latin verb meaning to project or stand out, and are often used this way in anatomy. Dictionaries also report thenar as a noun, true to the Greek word (for the whole palm) from which it is taken, but from my quick survey, it appears that today "thenar" is used mainly as an adjective, as in "the thenar crease", "a thenar muscle", "thenar pain", "a thenar pad", "thenar atrophy", and more.*
In common language, it's called the ball of the thumb, the fleshy part of the thumb, the meaty part of the thumb, the base of the thumb, the heel of the thumb, the butt of the thumb, the pulpy part of the thumb, or even the muscley part of the thumb. Since people so rarely talk about it, when people need to refer to it they invent a descriptive phrase on the fly, but often they invent the same phrase. The words "fleshy" and "meaty" suggest its consistency most clearly. People sometimes even independently invent the same word, "muscley". "Muscley" is much rarer than "thenar", and feels awkward and even childish, especially in writing, yet the average person could guess what you meant if you said "the muscley part of your thumb" and would have no idea what you meant if you said "your thenar eminence". (They might take the latter as a compliment.)
Some of these on-the-fly phrases are a little ambiguous, though, since more than one part of the hand on or near the thumb answers to the description, as you can see by googling for those phrases and seeing what people are talking about. This is why doctors and scientists prefer non-vernacular terms taken from Latin or Ancient Greek.
*Here's my educated guess about what happened (not fully checked out). As "thenar" is a noun in Greek, it was adopted as a noun in Renaissance Latin (e.g. in this translation of a 4th-century Greek medical encyclopedia, where it's defined as "the space between the index finger and the thumb"—not quite the modern meaning). The -ar ending makes it look like a Latin adjective (or an English adjective derived from Latin). It turns out that science and medicine have a lot of occasions for using "thenar" as an adjective, to distinguish various kinds of things: creases, muscles, homologous structures in other animals, etc. So, as the adjectival use gained prominence, its use as a noun atrophied. The noun use is not extinct today—see, for example, here—but its use as a noun seems to have declined starting around 1900, and today "thenar" seems to be most commonly found as a modifier on a noun, most frequently "eminence".
It's called Thenar
the soft part of the hand at the base of the thumb
The part you circled is the "base of the thumb", although I don't think of it as soft, more like "fleshy".
Edit: other answers have suggested "thenar", as a shortening of "thenar eminence". The underlying Greek word θέναρ just means the "palm of the hand", but I doubt anyone who is not either a palmist, a speaker of classical Greek, or a hand-surgeon would recognize the word.
Walt Whitman called it the "chuff of your hand" (in "Song of Myself").
This has always stuck with me, but a bit of searching seems to reveal that, surprisingly, nobody else has called it that.
I at least would have understood "chuff", but not "thenar" (well, not until I saw the other answers, anyway.)