Single quotes won’t interpolate anything, but double quotes will. For example: variables, backticks, certain escapes, etc.

Example:

$ echo "$(echo "upg")"
upg
$ echo '$(echo "upg")'
$(echo "upg")

The Bash manual has this to say:

3.1.2.2 Single Quotes

Enclosing characters in single quotes (') preserves the literal value of each character within the quotes. A single quote may not occur between single quotes, even when preceded by a backslash.

3.1.2.3 Double Quotes

Enclosing characters in double quotes (") preserves the literal value of all characters within the quotes, with the exception of $, `, , and, when history expansion is enabled, !. The characters $ and ` retain their special meaning within double quotes (see Shell Expansions). The backslash retains its special meaning only when followed by one of the following characters: $, `, ", , or newline. Within double quotes, backslashes that are followed by one of these characters are removed. Backslashes preceding characters without a special meaning are left unmodified. A double quote may be quoted within double quotes by preceding it with a backslash. If enabled, history expansion will be performed unless an ! appearing in double quotes is escaped using a backslash. The backslash preceding the ! is not removed.

The special parameters * and @ have special meaning when in double quotes (see Shell Parameter Expansion).

The accepted answer is great. I am making a table that helps in quick comprehension of the topic. The explanation involves a simple variable a as well as an indexed array arr.

If we set

a=apple      # a simple variable
arr=(apple)  # an indexed array with a single element

and then echo the expression in the second column, we would get the result / behavior shown in the third column. The fourth column explains the behavior.

See also:

• ANSI-C quoting with $'' – GNU Bash Manual
• Locale translation with $"" – GNU Bash Manual
• A three-point formula for quotes

If you’re referring to what happens when you echo something, the single quotes will literally echo what you have between them, while the double quotes will evaluate variables between them and output the value of the variable.

For example, this

#!/bin/sh
MYVAR=sometext
echo "double quotes gives you $MYVAR"
echo 'single quotes gives you $MYVAR'

will give this:

double quotes gives you sometext
single quotes gives you $MYVAR

Others explained it very well, and I just want to give something with simple examples.

Single quotes can be used around text to prevent the shell from interpreting any special characters. Dollar signs, spaces, ampersands, asterisks and other special characters are all ignored when enclosed within single quotes.

echo 'All sorts of things are ignored in single quotes, like $ & * ; |.'

It will give this:

All sorts of things are ignored in single quotes, like $ & * ; |.

The only thing that cannot be put within single quotes is a single quote.

Double quotes act similarly to single quotes, except double quotes still allow the shell to interpret dollar signs, back quotes and backslashes. It is already known that backslashes prevent a single special character from being interpreted. This can be useful within double quotes if a dollar sign needs to be used as text instead of for a variable. It also allows double quotes to be escaped so they are not interpreted as the end of a quoted string.

echo "Here's how we can use single ' and double " quotes within double quotes"

It will give this:

Here's how we can use single ' and double " quotes within double quotes

It may also be noticed that the apostrophe, which would otherwise be interpreted as the beginning of a quoted string, is ignored within double quotes. Variables, however, are interpreted and substituted with their values within double quotes.

echo "The current Oracle SID is $ORACLE_SID"

It will give this:

The current Oracle SID is test

Back quotes are wholly unlike single or double quotes. Instead of being used to prevent the interpretation of special characters, back quotes actually force the execution of the commands they enclose. After the enclosed commands are executed, their output is substituted in place of the back quotes in the original line. This will be clearer with an example.

today=`date '+%A, %B %d, %Y'`
echo $today

It will give this:

Monday, September 28, 2015

Since this is the de facto answer when dealing with quotes in Bash, I’ll add upon one more point missed in the answers above, when dealing with the arithmetic operators in the shell.

The Bash shell supports two ways to do arithmetic operation, one defined by the built-in let command and the other the $((..)) operator. The former evaluates an arithmetic expression while the latter is more of a compound statement.

It is important to understand that the arithmetic expression used with let undergoes word-splitting, pathname expansion just like any other shell commands. So proper quoting and escaping need to be done.

See this example when using let:

let 'foo = 2 + 1'
echo $foo
3

Using single quotes here is absolutely fine here, as there isn’t any need for variable expansions here. Consider a case of

bar=1
let 'foo = $bar + 1'

It would fail miserably, as the $bar under single quotes would not expand and needs to be double-quoted as

let 'foo = '"$bar"' + 1'

This should be one of the reasons, the $((..)) should always be considered over using let. Because inside it, the contents aren’t subject to word-splitting. The previous example using let can be simply written as

(( bar=1, foo = bar + 1 ))

Always remember to use $((..)) without single quotes

Though the $((..)) can be used with double quotes, there isn’t any purpose to it as the result of it cannot contain content that would need the double quote. Just ensure it is not single quoted.

printf '%dn' '$((1+1))'
-bash: printf: $((1+1)): invalid number
printf '%dn' $((1+1))
2
printf '%dn' "$((1+1))"
2

Maybe in some special cases of using the $((..)) operator inside a single quoted string, you need to interpolate quotes in a way that the operator either is left unquoted or under double quotes. E.g., consider a case, when you are tying to use the operator inside a curl statement to pass a counter every time a request is made, do

curl http://myurl.com --data-binary '{"requestCounter":'"$((reqcnt++))"'}'

Notice the use of nested double quotes inside, without which the literal string $((reqcnt++)) is passed to the requestCounter field.

There is a clear distinction between the usage of ' ' and " ".

When ' ' is used around anything, there is no “transformation or translation” done. It is printed as it is.

With " ", whatever it surrounds, is “translated or transformed” into its value.

By translation/ transformation I mean the following:
Anything within the single quotes will not be “translated” to their values. They will be taken as they are inside quotes. Example: a=23, then echo '$a' will produce $a on standard output. Whereas echo "$a" will produce 23 on standard output.

A minimal answer is needed for people to get going without spending a lot of time as I had to.

The following is, surprisingly (to those looking for an answer), a complete command:

$ echo ''

whose output is:

Backslashes, surprisingly to even long-time users of bash, do not have any meaning inside single quotes. Nor does anything else.