Learn Russian: Я анализирую таблицу с несколькими важными категориями. - I am analyzing a table with several important categories.

The preposition с (meaning with) requires the Instrumental case. In this sentence, несколько is in an oblique case (any case other than Nominative or Accusative). When this happens, it declines to match the case of the noun, becoming the Instrumental несколькими.

Doesn't несколько usually force the noun into the Genitive plural? Why are важными категориями in the Instrumental?

That Genitive plural rule only applies when несколько is in the Nominative or inanimate Accusative case. When a quantifier is in an oblique case like the Instrumental, it loses its special power. Instead, the quantifier, the adjective, and the noun all agree in the same case. Here, they are all Instrumental plural.

What declension pattern does несколько follow to become несколькими?

In oblique cases, quantifiers like сколько, столько, and несколько decline just like plural adjectives. The standard Instrumental plural ending for adjectives is -ыми or -ими. Here, it takes -ими, giving us несколькими.

I see анализирую ends in -ую. What is the infinitive and how does it conjugate?