How to create new data frame based on conditions

You could define a custom order for character strings by ordered(...) or factor(..., ordered = TRUE). In this way you could use >, <, (p)max/min to compare their priorities.

do.call(pmax,
  c(lapply(df, ordered, levels = c('cat', 'horse', 'dog', 'fish')), na.rm = TRUE)
)

# [1] fish dog  fish
# Levels: cat < horse < dog < fish

Data

df <- data.frame(Col_1 = c('cat', 'dog', 'cat'),
                 Col_2 = c('dog', 'horse', 'fish'),
                 Col_3 = c('horse', 'cat', NA),
                 Col_4 = c('fish', 'horse', NA))

Throwing my solution to the ring

library(dplyr)

df %>%
rowwise() %>%
  mutate(
    priority = intersect(c('fish', 'dog', 'horse', 'cat'), c(X1, X2, X3, X4))[1]
  ) %>%
  ungroup()

#> # A tibble: 3 × 5
#>   X1    X2    X3    X4    priority
#>   <chr> <chr> <chr> <chr> <chr>   
#> 1 cat   dog   horse fish  fish    
#> 2 dog   horse cat   horse dog     
#> 3 fish  fish  <NA>  <NA>  fish

^{Created on 2022-07-27 by the reprex package (v2.0.1)}

Convert the desired rows to vectors, and then take the first item from the intersect of the priority vector. If you only want the new column change mutate to transmute (or add ,.keep = 'none' to the mutate)

Another approach with base R with match and max.col:

prior_vec <- c('cat', 'horse', 'dog', 'fish')

s <- sapply(df, match, table = prior_vec, nomatch = 0)

df[cbind(1:nrow(df), max.col(s))]

which gives:

# [1] "fish" "dog"  "fish"

You can make your data into a long dataset with row numbers. Then you can arrange it as per your preference after grouping based on the row numbers. Then you can use slice to get the desired data frame.

library(tidyverse)

tribble(~col1, ~col2, ~col3, ~col4,
        "cat", "dog", "horse", "fish",
        "dog", "horse", "cat", "horse",
        "cat", "fish", NA, NA) -> df


df %>% 
  mutate(rn = row_number()) %>% 
  pivot_longer(-rn) %>% 
  drop_na() %>% 
  group_by(rn) %>% 
  arrange(match(value, c("fish", "dog", "horse", "cat")), .by_group = T) %>% 
  slice(1) %>% 
  ungroup() %>% 
  select(value)

# A tibble: 3 × 1
  value
  <chr>
1 fish 
2 dog  
3 fish