Answer 1

You could also use indexOf to check the content of array. Return -1 if not exist, else variable exists in the array.
var hi = prompt("Please type in a way to greet someone");
var grt = ["Hi", "Hey", "Hello"];
if(grt.indexOf(hi)!=-1){
alert("Hello!");
}

Answer 2

You can't compare like that. hi must be checked against each element in the array.
var hi = prompt("Please type in a way to greet someone");
var grt = ["Hi", "Hey", "Hello"];
for (var i = 0; i < grt.length; i++) {
if(hi == grt[i]){
alert("Hello!");
}
}