Learn French: Je ne comprends nullement cette insinuation. - I do not understand this insinuation in the least.

Nullement is a strong negative adverb meaning "not at all" or "in no way." It carries the exact same meaning as pas du tout, but it belongs to a much more formal or literary register. You will usually encounter it in formal writing, formal speeches, or when someone is being polite but very firm.

Why is there no pas in this negative sentence?

Just like jamais (never) or rien (nothing), nullement is a negative word that completely replaces pas. Saying je ne comprends pas nullement would be grammatically incorrect. The standard negative structure is simply ne + verb + nullement.

How can I easily remember that insinuation takes the feminine cette?

The word insinuation ends in -tion. In French, almost all nouns ending in -tion are feminine. Because it is feminine and singular, it requires the feminine demonstrative adjective cette (this or that) instead of the masculine ce or cet.