One option:
=COUNTIF(B:B; "Mammal") + COUNTIF(B:B; "Bird")
According to the documentation:
Notes
COUNTIF can only perform conditional counts with a single criterion.
To use multiple criteria, use COUNTIFS or the database functions
DCOUNT or DCOUNTA.
COUNTIFS: This function is only available in the new Google Sheets.
Example:
=DCOUNTA(B:B; 2; {"Type"; "Mammal"; "Bird"})