How do I check if a variable exists in an 'if' statement?

In modern bash (version 4.2 and above):

[[ -v name_of_var ]]

From help test:

-v VAR, True if the shell variable VAR is set

Depends what you mean by exists.

Does a variable that has been declared but not assigned exist?

Does an array (or hash) variable that has been assigned an empty list exist?

Does a nameref variable pointing to a variable that currently isn't assigned exist?

Do you consider $-, $#, $1 variables? (POSIX doesn't).

In Bourne-like shells, the canonical way is:

if [ -n "${var+set}" ]; then
  echo '$var was set'
fi

That works for scalar variables and other parameters to tell if a variable has been assigned a value (empty or not, automatically, from the environment, assigments, read, for or other).

For shells that have a typeset or declare command, that would not report as set the variables that have been declared but not assigned (note that in zsh, declaring a variable assigns a value, a default one if not specified).

For shells that support arrays, except for yash and zsh that would not report as set array variables unless the element of indice 0 has been set.

For bash (but not ksh93 nor zsh), for variables of type associative array, that would not report them as set unless their element of key "0" has been set.

For ksh93 and bash, for variables of type nameref, that only returns true if the variable referenced by the nameref is itself considered set.

For ksh, zsh and bash, a potentially better approach could be:

if ((${#var[@]})); then
  echo '$var (or the variable it references for namerefs) or any of its elements for array/hashes has been set'
fi

For ksh93, zsh and bash 4.4 or above, there's also:

if typeset -p var 2> /dev/null | grep -q '^'; then
  echo '$var exists'
fi

Which will report variables that have been set or declared.

As mentioned in the answer on SO, here is a way to check:

if [ -z ${somevar+x} ]; then echo "somevar is unset"; else echo "somevar is set to '$somevar'"; fi

where ${somevar+x} is a parameter expansion which evaluates to the null if var is unset and substitutes the string "x" otherwise.

Using -n, as suggested by the other answer, will only check if the variable contains empty string. It will not check its existence.