How to create and add contact form on page in WordPress?

Step 1. Install and activate Contact Form 7 plugin.
Step 2. Add new form depends on your needs (WP Dashboard > Contact > Add New). You can also visit http://contactform7.com/getting-started-with-contact-form-7/ if you need more details about creating a new form.

Step 3. Copy the shortcode from the created form (see the screenshot above) and then add this shortcode to the choosen page.

Update the page and view the form on the page.

Notes: If you are using Visual Composer then you can add the shortcode inside text block element.