I have a function that logs in a user successfully.

_login: function() {
   var email = this.$.emailvalue.value;
   var password = this.$.passwordvalue.value;

   return this.$.authenticate.signInWithEmailAndPassword(email, password).then(function() {
      // Sign-in successful.
      //this._animateView(); **returns undefined**
    }, function(error) {
      // An error happened.
    //     // Handle Errors here.
        var errorCode = error.code;
        var errorMessage = error.message; 
        console.log(errorMessage);
        // this._animateErrorView(); **returns undefined**
    });

  },

What I want is to call a function this._animateView(); when a user logs in successfully and this._animateErrorView() when there is an error.

If I try to do that It returns undefined.. How do I solve this issue

The meaning of this is different in the callback function.

You can fix this in many ways, but here are two.

capture the value of this in another variable

You cannot prevent this from getting a new value, since that's simply how JavaScript closures work. But what you can do is define a different value with the value that you need. A idiomatic name for the variable is self, although I personally prefer something more descriptive:

_login: function() {
   var email = this.$.emailvalue.value;
   var password = this.$.passwordvalue.value;
   var self = this;    
   return this.$.authenticate.signInWithEmailAndPassword(email, password).then(function() {
        self._animateView();
    }, function(error) {
        var errorCode = error.code;
        var errorMessage = error.message; 
        console.log(errorMessage);
        self._animateErrorView();
    });
  }

use fat arrow functions

ES6 defines a new way to declare functions with so-called fat arrows (=>). Aside from being slightly less code, these ensure the value of this in the lambda/callback remains unchanged. So

_login: function() {
   var email = this.$.emailvalue.value;
   var password = this.$.passwordvalue.value;
   return this.$.authenticate.signInWithEmailAndPassword(email, password).then(() => {
        this._animateView();
    }, (error) => {
        var errorCode = error.code;
        var errorMessage = error.message; 
        console.log(errorMessage);
        this._animateErrorView();
    });
  }