List.contains() fails while .equals() works

Just because your Test‘s equals may return true when you pass a String to it doesn’t mean that String‘s equals will ever return true when you pass a Test instance to it. In fact, String‘s equals can only return true when the instance passed to it is another String :

public boolean equals(Object anObject) {
    if (this == anObject) {
        return true;
    }
    if (anObject instanceof String) { // the passed instance must be a String
        String anotherString = (String)anObject;
        int n = value.length;
        if (n == anotherString.value.length) {
            char v1[] = value;
            char v2[] = anotherString.value;
            int i = 0;
            while (n-- != 0) {
                if (v1[i] != v2[i])
                    return false;
                i++;
            }
            return true;
        }
    }
    return false;
}

ArrayList‘s contains calls indexOf which uses the equals method of the searched instance (the String “a” in your example), not the element type of the List (which is Test in your case) :

public int indexOf(Object o) {
    if (o == null) {
        for (int i = 0; i < size; i++)
            if (elementData[i]==null)
                return i;
    } else {
        for (int i = 0; i < size; i++)
            if (o.equals(elementData[i])) // o in your case is a String while
                                          // elementData[i] is a Test
                                          // so String's equals returns false
                return i;
    }
    return -1;
}