Adding one record example:

<?php
$item = array(
  'field_338' => 'Test Ticket', //338 - Text field ID
  'field_426' => '2017-12-29', //426 - Field ID of the type "Date with calendar"
  'field_429' => '166,167', //429 - Field ID of the "Dropdown multiselect"
);

$params = array(
  'key' => 'XgDXFsTbNRkMpRq81bBrmRAf56i5oS0oN9bp4jLH', //API key
  'username' => 'admin', //username
  'password' => 'admin', //password
  'action' => 'insert', //action
  'entity_id' => 34, //ID of the entity to which the record will be added
  'items' => $item, //array of records
);
 						                                    
$ch = curl_init('http://localhost/rukovoditel/api/rest.php'); //API Url
curl_setopt($ch, CURLOPT_HEADER, false);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_POSTFIELDS, http_build_query($params));
curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, false);
curl_setopt($ch, CURLOPT_TIMEOUT, 10);			
$result = curl_exec($ch);
curl_close($ch);

if($result)
{
  $result = json_decode($result,true);
  
  print_r($result);
}

If successful, returns the ID of the added records:

Array
(
    [status] => success
    [data] => Array
        (
            [id] => 119
        )
)

In case of an error, returns the text of the error:

Array
(
    [status] => error
    [error_code] => 
    [error_message] => field_429 not exist in entity 34
)
Field Description
key API key. Generate on the page: Extension - Tools - API.
username The username who has the access to perform the action.
password The user's password.
action The action to be performed.
entity_id The ID of the entity for which the action is performed.
items An array of records, including the fields of the record.
  parent_item_id - an optional field is required if the entry is added to the parent record.
 

field_X - the field to add, where X is the field ID. The field ID can be found on the "Field Configuration" page. This field takes a value as a string.

Date
For the "date" field type, use the format YYYY-MM-DD.

Drop-down list
For the field type "Drop-down list" use the option ID, which can be found on the "Options" page for each field. If you are using a Global List, use the option ID from the Global List. If there is a list with multiple values, enter them separated by commas.

Attachments
To transfer attachments via the API, specify an http link to the file in the field value. You can add multiple urls separated by commas.

Example of adding multiple records:

$items = [];

$items[] = array( 
  'field_338' => 'Test Ticket 1', //338 - Text field ID
  'field_426' => '2017-12-29', //426 - Field ID of the type "Date with calendar"
  'field_429' => '166,167', //429 - Field ID of the "Dropdown multiselect"
);

$items[] = array(
  'field_338' => 'Test Ticket 2', //338 - Text field ID
  'field_426' => '2017-12-29', //426 - Field ID of the type "Date with calendar"
  'field_429' => '166,167', //429 - Field ID of the "Dropdown multiselect"
);

$params = array(
  'key' => 'XgDXFsTbNRkMpRq81bBrmRAf56i5oS0oN9bp4jLH', //API key
  'username' => 'admin', //username
  'password' => 'admin', //password
  'action' => 'insert', //action
  'entity_id' => 34, //ID of the entity to which the record will be added
  'items' => $items, //array of records
);