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Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions)

My question is that what does the conjugacy mean here precisely? What do the mutual and the self mean in terms of the conjugacy?

See Wikipedia page:

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Let the spacetime metric have signature $(s,t)$.

  1. Mutual conjugated means that the left and right Weyl representations are each other's complex conjugate complex representations. This happens if $s-t~=~2\mod 4$.

  2. Self-conjugated means that the left and right Weyl representations are (pseudo) real representations. This happens if $s-t~=~0\mod 4$.

(Weyl representations only exist in even spacetime dimensions $s+t~=~0\mod 2$.)

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