The LaTeX command for exists (∃) is \exists (there exists / existential quantifier).
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Write \exists inside math mode, that is between $…$ for inline math or in a \[ … \] or equation block for display math. It is part of core LaTeX math, so no extra package is required.
\exists keeps its spacing inside a formula; for a word such as "and" in the middle of one, use \text{ and } from amsmath. With unicode-math under XeLaTeX or LuaLaTeX, typing ∃ in a formula gives the same symbol.
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Use \exists in math mode; it needs no extra package. For example, $p \exists q$.
None: \exists is built into LaTeX, so no extra package is required.