Edit in JSFiddle

new Vue({
  el: '#app',
  data: {
    newTag: '',
    //tags: ['Vue', 'React', 'Jquery']
    tags: []
  },

  methods: {
    add(tag) {
        if (tag && this.tags.indexOf(tag) === -1) {
          this.tags.push(tag)
        }
        this.newTag = ''

      },
      del(index) {
        this.tags.splice(index, 1)

      },
      delLastTag() {
        if (this.newTag) {
          return
        }
        this.tags.pop()
      }
  }
});

<div id="app">
  <div class="tags-wrap">
    <div class="tags">
      <span class="content" v-for="(tag, index) in tags" :key="index">{{tag}}
      <span class="del" @click="del(index)">&times;</span>
      </span>
      <input class="tags-input" type="text" v-model="newTag" v-on:keyup.delete.stop="delLastTag()" v-on:keyup.enter="add(newTag)" placeholder="click enter">
    </div>

  </div>
  <br>
  <div class="form-group">
    <label for="">Tags:</label>
    <code>{{tags}}</code>
  </div>
</div>

.tags-wrap {
  background-color: #fff;
  border: 1px solid #ccc;
  overflow: hidden;
  padding-left: 4px;
  padding-top: 4px;
  cursor: text;
  text-align: left;
  -webkit-appearance: textfield;
}

.tags-wrap::after {
  content: "";
  display: block;
  height: 0;
  clear: both;
}

.tags-input {
  background: transparent;
  border: 0;
  color: #777;
  font-size: 13px;
  font-weight: 400;
  margin-bottom: 6px;
  margin-top: 1px;
  outline: none;
  padding: 4px;
  padding-left: 0;
  width: 80px
}

.content {
  background-color: #e0e0e0;
  border-radius: 8px;
  color: rgba(0, 0, 0, .87);
  line-height: 25px;
  display: inline-block;
  font-size: 14px;
  font-weight: 400;
  margin-bottom: 4px;
  margin-right: 4px;
  padding: 6px;
}

.del {
  cursor: pointer;
  font-weight: 700;
  color: #1F1F1F;
}