How to paste a string on each element of a vector of strings using apply in R?

No need for apply(), just use paste():

R> d <- c("Mon","Tues","Wednes","Thurs","Fri","Satur","Sun")
R> week <- paste(d, "day", sep="")
R> week
[1] "Monday"    "Tuesday"   "Wednesday" "Thursday"  
[4] "Friday"    "Saturday"  "Sunday"   
R> 

Other have already indicated that since paste is vectorised, there is no need to use apply in this case.

However, to answer your question: apply is used for an array or data.frame. When you want to apply a function over a list (or a vector) then use lapply or sapply (a variant of lapply that simplifies the results):

sapply(d, paste, "day", sep="")
        Mon        Tues      Wednes       Thurs         Fri       Satur 
   "Monday"   "Tuesday" "Wednesday"  "Thursday"    "Friday"  "Saturday" 
        Sun 
   "Sunday" 

Apart from paste/paste0 there are variety of ways in which we can add a string to every element in the vector.

1) Using sprintf

sprintf("%sday", d)
#[1] "Monday"    "Tuesday" "Wednesday" "Thursday"  "Friday"  "Saturday"  "Sunday" 

2) glue

glue::glue("{d}days")

Here {d} is evaluated as R code. This can be wrapped in as.character if needed.

3) str_c in stringr

stringr::str_c(d, "day")

whose equivalent is

4) stri_c in stringi

stringi::stri_c(d, "day")

5) stringi also has stri_paste

stringi::stri_paste(d, "day")