Arrays can only be compared with Arrays.equals().

You probably want an ArrayList of ArrayLists.

ArrayList<ArrayList<Integer>> j = new ArrayList<ArrayList<Integer>>();
ArrayList<Integer> w = new ArrayList<Integer>();
w.add(1); w.add(2);
j.add(w);
ArrayList<Integer> t = new ArrayList<Integer>();
t.add(1); t.add(2);
return j.contains(t); // should return true.

You are right. ArrayList.contains() tests equals(), not object identity:

returns true if and only if this list contains at least one element e such that (o==null ? e==null : o.equals(e))

If you got a NullPointerException, verify that you initialized your list, either in a constructor or the declaration. For example:

private List<String> rssFeedURLs = new ArrayList<String>();

Yes. You need to define your NimPlayer class like this (if you want to compare two NimPlayer objects only by their userName)

package com.example.schooltimetable;

import java.util.Objects;

public class NimPlayer {
  private String userName;
  private String familyName;
  private String givenName;
  private int gamesWon;
  private int gamesPlayed;

  public String getUserName() {
    return userName;
  }

  public void setUserName(String userName) {
    this.userName = userName;
  }

  public String getFamilyName() {
    return familyName;
  }

  public void setFamilyName(String familyName) {
    this.familyName = familyName;
  }

  public String getGivenName() {
    return givenName;
  }

  public void setGivenName(String givenName) {
    this.givenName = givenName;
  }

  public int getGamesWon() {
    return gamesWon;
  }

  public void setGamesWon(int gamesWon) {
    this.gamesWon = gamesWon;
  }

  public int getGamesPlayed() {
    return gamesPlayed;
  }

  public void setGamesPlayed(int gamesPlayed) {
    this.gamesPlayed = gamesPlayed;
  }

  @Override
  public boolean equals(Object o) {
    if (this == o)
      return true;
    if (o == null || getClass() != o.getClass())
      return false;
    NimPlayer nimPlayer = (NimPlayer) o;
    return Objects.equals(userName, nimPlayer.userName);
  }

  @Override
  public int hashCode() {
    return Objects.hash(userName);
  }
}

Now to check if the arraylist contains :

public void showCertainPlayerInformation(ArrayList<NimPlayer> playerList, String inputName)
    {
      NimPlayer n = new NimPlayer();
      n.setUserName(inputName);
      if(playerList.contains(n)){
          ....
          return;
      }
      System.out.println("The player does not exist");
     }