new Vue({
  el:"#app",
  data:{
    foo: "bar"
  },
  methods:{
    doSomethingAsynchronous(){
      setTimeout(function(){
        // This is wrong! Inside this function,
        // "this" refers to the window object.
        this.foo = "baz";
      }, 1000);
    }
  }
})

new Vue({
  el:"#star-wars-people",
  data:{
    people: null
  },
  mounted: function(){
    $.getJSON("http://swapi.co/api/people/", function(data){
      // Again, this is wrong! "this", here, refers to the window.
      this.people = data.results;
    })
  }
})

You can capture the correct this using a closure.

new Vue({
  el:"#star-wars-people",
  data:{
    people: null
  },
  mounted: function(){
    // Before executing the web service call, save this to a local variable
    var self = this;
    $.getJSON("http://swapi.co/api/people/", function(data){
      // Inside this call back, because of the closure, self will
      // be accessible and refers to the Vue object.
      self.people = data.results;
    })
  }
})

You can bind the callback function.

new Vue({
  el:"#star-wars-people",
  data:{
    people: null
  },
  mounted:function(){
    $.getJSON("http://swapi.co/api/people/", function(data){
      this.people = data.results;
    }.bind(this));
  }
})

new Vue({
  el:"#star-wars-people",
  data:{
    people: null
  },
  mounted: function(){
    $.getJSON("http://swapi.co/api/people/", data => this.people = data.results);
  }
})

Caution! Arrow functions are a syntax introduced in Ecmascript 2015. It is not yet supported but all modern browsers, so only use it if you are targetting a browser you know supports it, or if you are compiling your javascript down to ES5 syntax using something like babel.

new Vue({
  el:"#app",
  data:{
    foo: "bar"
  },
  methods:{
    // This is wrong! Arrow functions capture "this" lexically
    // and "this" will refer to the window.
    doSomething: () => this.foo = "baz"
  }
})

new Vue({
  el:"#app",
  data:{
    foo: "bar"
  },
  methods:{
    doSomething: function(){
      this.foo = "baz"
    }
  }
})

Alternatively, if you are using a javascript compiler or a browser that supports Ecmascript 2015

new Vue({
  el:"#app",
  data:{
    foo: "bar"
  },
  methods:{
    doSomething(){
      this.foo = "baz"
    }
  }
})