If the string is already stored in a variable you can use bash's parameter expansion, specifially ${parameter,,pattern} (available since bash 4.0), where parameter is the name of your variable and pattern is ommitted:

$ string="Hello, World!"

$ echo $string

Hello, World!

$ echo ${string,,}

hello, world!

Note that this does not change the value of the variable, only the output. To change the variable you have to assign the new value:

$ echo $string

Hello, World!

$ string=${string,,}

$ echo $string

hello, world!

The upper-case conversion works with ${parameter^^pattern}:

$ echo ${string^^}

HELLO, WORLD!

This works also with Unicode strings (at least with current bash versions, probably needs at least bash 4.3):

$ string='ἈΛΦΆβητος'

$ echo ${string,,}

ἀλφάβητος

$ echo ${string^^}

ἈΛΦΆΒΗΤΟΣ

If you are using zsh, you can use Parameter Expansion Flags (${(FLAGS)NAME}; available since zsh 2.5) to achieve the same results. The bash syntax does not work in zsh ¹⁾. The flag for lower case is L; for upper case it is U:

$ string="Hello, World!"

$ echo ${(L)string}

hello, world!

$ echo ${(U)string}

HELLO, WORLD!

$ echo $string

Hello, World!"

This also works with Unicode strings (at least since zsh 5.0; I did not try with earlier versions):

$ string='ἈΛΦΆβητος'

$ echo ${(L)string} 

ἀλφάβητος

$ echo ${(U)string}  

ἈΛΦΆΒΗΤΟΣ

¹⁾ Although, seeing that zsh had this for far longer, it should probably be: "the zsh syntax does not work in bash.