You may try pmax
df$c <- pmax(df$a, df$b)
df
# a b c
# 1 dog <NA> dog
# 2 mouse <NA> mouse
# 3 <NA> cat cat
# 4 bird <NA> bird
…or ifelse
:
df$c <- ifelse(is.na(df$a), df$b, df$a)
For more general solutions in cases with more than two columns, you find several ways to implement coalesce in R here.