Basically, props and state are two ways the component can know what and how to render. Which part of the application state belongs to state and which to some top-level store, is more related to your app design, than to how React works. The simplest way to decide, IMO, is to think, whether this particular piece of data is useful for application as a whole, or it’s some local information. Also, it’s important to not duplicate state, so if some piece of data can be calculated from props – it should calculated from props.

For example, let’s say you have some dropdown control (which wraps standart HTML select for custom styling), which can a) select some value from list, and b) be opened or closed (i.e., the options list displayed or hidden).
Now, let’s say your app displays a list of items of some sort and your dropdown controls filter for list entries. Then, it would be best to pass active filter value as a prop, and keep opened/closed state local. Also, to make it functional, you would pass an onChange handler from parent component, which would be called inside dropdown element and send updated information (new selected filter) to the store immediately. On the other hand, opened/closed state can be kept inside dropdown component, because the rest of the application doesn’t really care if the control is opened, until user actually changes it value.

The following code is not completely working, it needs css and handling dropdown click/blur/change events, but I wanted to keep example minimal. Hope it helps to understand the difference.

const _store = {
    items: [
    { id: 1, label: 'One' },
    { id: 2, label: 'Two' },
    { id: 3, label: 'Three', new: true },
    { id: 4, label: 'Four', new: true },
    { id: 5, label: 'Five', important: true },
    { id: 6, label: 'Six' },
    { id: 7, label: 'Seven', important: true },
    ],
  activeFilter: 'important',
  possibleFilters: [
    { key: 'all', label: 'All' },
    { key: 'new', label: 'New' },
    { key: 'important', label: 'Important' }
  ]
}

function getFilteredItems(items, filter) {
    switch (filter) {
    case 'all':
        return items;

    case 'new':
        return items.filter(function(item) { return Boolean(item.new); });

    case 'important':
        return items.filter(function(item) { return Boolean(item.important); });

    default:
        return items;
  }
}

const App = React.createClass({
  render: function() {
    return (
            <div>
            My list:

            <ItemList   items={this.props.listItems} />
          <div>
            <Dropdown 
              onFilterChange={function(e) {
                _store.activeFilter = e.currentTarget.value;
                console.log(_store); // in real life, some action would be dispatched here
              }}
              filterOptions={this.props.filterOptions}
              value={this.props.activeFilter}
              />
          </div>
        </div>
      );
  }
});

const ItemList = React.createClass({
  render: function() {
    return (
      <div>
        {this.props.items.map(function(item) {
          return <div key={item.id}>{item.id}: {item.label}</div>;
        })}
      </div>
    );
  }
});

const Dropdown = React.createClass({
    getInitialState: function() {
    return {
        isOpen: false
    };
  },

  render: function() {
    return (
        <div>
            <select 
            className="hidden-select" 
          onChange={this.props.onFilterChange}
          value={this.props.value}>
            {this.props.filterOptions.map(function(option) {
            return <option value={option.key} key={option.key}>{option.label}</option>
          })}
        </select>

        <div className={'custom-select' + (this.state.isOpen ? ' open' : '')} onClick={this.onClick}>
            <div className="selected-value">{this.props.activeFilter}</div>
          {this.props.filterOptions.map(function(option) {
            return <div data-value={option.key} key={option.key}>{option.label}</div>
          })}
        </div>
      </div>
    );
  },

  onClick: function(e) {
    this.setState({
        isOpen: !this.state.isOpen
    });
  }
});

ReactDOM.render(
  <App 
    listItems={getFilteredItems(_store.items, _store.activeFilter)} 
    filterOptions={_store.possibleFilters}
    activeFilter={_store.activeFilter}
    />,
  document.getElementById('root')
);