If you want to show today's member birthdays, you can use
<!-- BEGIN BIRTHDAYS -->
<a href="{MEMBERPROFILEURL}" class="{MEMBERGROUPNAMESTYLE}">{MEMBERNAME}</a><IF {MEMBERAGE}> ({MEMBERAGE})</IF>,
<!-- END BIRTHDAYS -->
in any template. This is cached for speed.<a href="{MEMBERPROFILEURL}" class="{MEMBERGROUPNAMESTYLE}">{MEMBERNAME}</a><IF {MEMBERAGE}> ({MEMBERAGE})</IF>,
<!-- END BIRTHDAYS -->
If you have the Calendar and Calendar birthdays switches enabled, the calendar will show upcoming member birthdays on it. At Admin Panel -> Settings -> General you can set how recently the member needs to have logged in for their birthday to be listed. If you have a very large number of members (10,000+) you may notice a slowdown in viewing the calendar, and can speed things up by setting it to show fewer birthdays.
As a way of keeping people engaged with your site, a happy birthday email is automatically sent to each user on their birthday. If you prefer not to send this email, turn it off at Admin -> E-mails -> E-mail Preferences -> "Send email to user on their birthday?". If you'd like to customize the email text, go to Admin -> Customizations -> Language and search for language items where the name contains birthday (email_birthdaysubject is the subject line and email_birthdaybody is the body text).