How do I set a variable to the output of a command in Bash?

$(sudo run command)

---

If you're going to use an apostrophe, you need `, not '. This character is called "backticks" (or "grave accent"):

#!/bin/bash

VAR1="$1"
VAR2="$2"

MOREF=`sudo run command against "$VAR1" | grep name | cut -c7-`

echo "$MOREF"

In addition to backticks `command`, command substitution can be done with $(command) or "$(command)", which I find easier to read, and allows for nesting.

OUTPUT=$(ls -1)
echo "${OUTPUT}"

MULTILINE=$(ls \
   -1)
echo "${MULTILINE}"

Quoting (") does matter to preserve multi-line variable values; it is optional on the right-hand side of an assignment, as word splitting is not performed, so OUTPUT=$(ls -1) would work fine.