Variables inside single quotes ' don't get expanded by POSIX compatible shells, so using a shell variable in a sed substitution requires the use of double quotes " instead of single quotes ':

$ var="he"
$ echo "hello" | sed "s/$var/XX/"
XXllo

$ var="he"
$ echo "hello" | sed 's/$var/XX/'
hello

Be careful of command injection when evaluating variables:

$ var='./&/;x;w/etc/passwd
> x;s/he'
$ echo "hello" | sed "s/$var/XX/"
sed: /etc/passwd: Permission denied

If the above was run as root the output would have been indistinguishable from the first example, and the contents of /etc/passwd would be destroyed.

Using escaped brackets, you can define a capturing group in a pattern that can be backreferenced in the substitution string with \1:

$ echo Hello world! | sed 's/\(Hello\) world!/\1 sed/'
Hello sed

With multiple groups:

$ echo one two three | sed 's/\(one\) \(two\) \(three\)/\3 \2 \1/'
three two one

$ echo one two three | sed -E 's/(one) (two) (three)/\3 \2 \1/'
three two one

Words consisting of letter, digits and underscores can be matched using the expression [[:alnum:]_]\{1,\}:

$ echo Hello 123 reg_exp | sed 's/\([[:alnum:]_]\{1,\}\) \([[:alnum:]_]\{1,\}\) \([[:alnum:]_]\{1,\}\)/\3 \2 \1/'
reg_exp 123 Hello

$ echo Hello 123 reg_exp | sed 's/\(\w\w*\) \(\w\w*\) \(\w\w*\)/\3 \2 \1/'
reg_exp 123 Hello

POSIX/IEEE Open Group Base Specification says:

[2addr] s/BRE/replacement/flags

Substitute the replacement string for instances of the BRE in the pattern space. Any character other than backslash or newline can be used instead of a slash to delimit the BRE and the replacement. Within the BRE and the replacement, the BRE delimiter itself can be used as a literal character if it is preceded by a backslash.

There are cases when the delimiter / for sed replacement is in the BRE or replacement, triggering errors like:

$ echo "2/3/4" | sed "s/2/3/X/"
sed: -e expression #1, char 7: unknown option to `s'

For this, we can use different delimiters such as # or _ or even a space:

$ echo "2/3/4" | sed "s#2/3#X#"
X/4
$ echo "2/3/4" | sed "s_2/3_X_"
X/4
$ echo "2/3/4" | sed "s 2/3 X "
X/4

If we want to replace only the first occurrence in a line, we use sed as usual:

$ cat example
aaaaabbbbb
aaaaaccccc
aaaaaddddd
$ sed 's/a/x/' example
xaaaabbbbb
xaaaaccccc
xaaaaddddd

But what if we want to replace all occurrences?

We just add the g pattern flag at the end:

$ sed 's/a/x/g' example
xxxxxbbbbb
xxxxxccccc
xxxxxddddd

And if we want to replace one specific occurrence, we can actually specify which one:

$ sed 's/a/x/3' example
aaxaabbbbb
aaxaaccccc
aaxaaddddd

/3 being the 3rd occurrence.

$ sed 's/b/y/2g' example
aaaaabyyyy
aaaaaccccc
aaaaaddddd

$ sed 's/c/z/g3' example
aaaaabbbbb
aaaaacczzz
aaaaaddddd