When are square brackets required in a Bash if statement?

Usually, I use square brackets in the if statement:

if [ "$name" = 'Bob' ]; then ...

But, when I check if grep succeeded I don't use the square brackets:

if grep -q "$text" $file ; then ...

When are the square brackets necessary in the if statement?

The square brackets are a synonym for the test command. An if statement checks the exit status of a command in order to decide which branch to take. grep -q "$text" is a command, but "$name" = 'Bob' is not--it's just an expression. test is a command, which takes an expression and evaluates it:

if test "$name" = 'Bob'; then ...

Since square brackets are a synonym for the test command, you can then rewrite it as your original statement:

if [ "$name" = 'Bob' ]; then ...

[ is actually a command, equivalent (almost, see below) to the test command. It's not part of the shell syntax. (Both [ and test, depending on the shell, are often built-in commands as well, but that doesn't affect their behavior, except perhaps for performance.)

An if statement executes a command and executes the then part if the command succeeds, or the else part (if any) if it fails. (A command succeeds if it exits with a status ($?) of 0, fails if it exits with a non-zero status.)

In

if [ "$name" = 'Bob' ]; then ...

the command is

[ "$name" = 'Bob' ]

(You could execute that same command directly, without the if.)

In

if grep -q "$text" $file ; then ...

the command is

grep -q "$text" $file

man [ or man test for more information.

FOOTNOTE: Well, the [ command is almost equivalent to the test command. The difference is that [ requires ] as its last argument, and test does not -- and in fact doesn't allow it (more precisely, test doesn't treat a ] argument specially; for example it could be a valid file name). (It didn't have to be implemented that way, but a [ without a matching ] would have made a lot of people very very nervous.)

The best way to think of the [ ... ] syntax, is to consider [ to be a program - which it is!

Check this out:

~ $ ls /usr/bin/\[ 
/usr/bin/[

on the other hand, you're probably not using that version of it since bash also provides [ as a shell built-in.

Anyway, to answer your question: What if does is run the command you give it and see it the return value is 0 or not. You use [ to do other, more interesting comparisons such as string comparisons. See man [ and man bash.