First thing I see off the bat is that you're double declaring Bernice -- you only need to type string once, when you're initially declaring the variable. After that you can just use it normally. i.e.
1 2
string Bernice;
Bernice = "Bernice";
Second thing is you're accepting input into an integer and then comparing it to a string which makes no sense. Fix those things and you might be able to get it working.
1 #include<iostream>
2 usingnamespace std;
3
4 int main()
5 {
6 int name; // This needs to be a string
7 string Bernice; // You do not need this, the next line covers the whole declaration
8 string Bernice= "Bernice";
9
10 cout << "Please enter a name" << endl;
11 cin>> name;
12
13 while (name!=Bernice)
14 {
15 cout << "Please enter another name" << endl;
16 cin >> name;
17 }
18
19 cout << "Correct name, You entered Bernice" << endl;
20
21 }
22