Edit in JSFiddle

var Example = React.createClass({

    getInitialState: function(){
        return ({
            hashtag: 'nofilter',
            mediacount: 0
        })
    },
    
    onUpdate: function(val){
      this.setState({
          hashtag: val
      });
    },
    
    render: function() {
        return (
          <div>
              <HashTagInput onUpdate={this.onUpdate} hashtag={this.state.hashtag}  />
              <HashTagCount hashtag={this.state.hashtag} />
          </div>
        )
    }
    
});

var HashTagInput = React.createClass({

    countHashTag: function() {
      var newHashTag = this.refs.inputHashTag.getDOMNode().value;
      this.props.onUpdate(newHashTag);
    },
    
    render: function () {
      return ( 
          <div>
            <input id="inputHashTag" ref="inputHashTag" type="text" placeholder="Insert a hashtag" defaultValue={this.props.hashtag}/>
            <button onClick={this.countHashTag} >Count</button>
          </div>
      );
    }
});

var HashTagCount = React.createClass({
    
    getInitialState: function(){
        return ({
            mediacount: (this.props.mediacount ? this.props.mediacount : 0)
        })
    },

    componentDidMount: function () {
        this.getHashTagCount(this.props.hashtag);
    },
    
    componentWillReceiveProps: function(props){
        this.getHashTagCount(props.hashtag);
    },
    
    getHashTagCount: function(hashtag) {
    
        $.ajax({
            type: "GET",
            "method": "GET",
            url: 'https://api.instagram.com/v1/tags/' + hashtag + '?client_id=06900e749be644729a42b3857dc07822',
            dataType: "jsonp",
            success: function (result) {
                var count = result.data.media_count
                this.setState({
                  hashtag: result.data.name,
                  mediacount: count.toString().replace(/\B(?=(\d{3})+(?!\d))/g, ",")
                });
            }.bind(this)
        });
    
    },

    render: function () {
        return ( < div > 
            {this.state.mediacount} tagged with <em>{this.state.hashtag}</em>
      </div>
    );
  }
});

React.render(<Example />, document.getElementById('input'));

<script src="https://facebook.github.io/react/js/jsfiddle-integration.js"></script>

<div id="input"></div>

body {
    font-family:Arial;
}

div {
    clear:both;
    width:100%;
    text-align:center;
    margin-top:2em;
}

div > div {
    clear:both;
}

input[type=text] {
    width:70%;
    float:left;
}

button[type=submit] {
    width:20%;
    float:right;
}