Use function wp_handle_upload()
The documentation is an example:
https://codex.wordpress.org/Function_Reference/wp_handle_upload
Example:
<form class="user_upload_photo none" enctype="multipart/form-data" action="" method="POST">
<?php wp_nonce_field('my_file_upload', 'fileup_nonce'); ?>
<input class="file_input_text" name="my_file_upload" type="file" />
<input class="btn" type="submit" value="Загрузить файл" />
</form>
handler:
if (wp_verify_nonce($_POST['fileup_nonce'], 'my_file_upload')) {
if (!function_exists('wp_handle_upload'))
require_once( ABSPATH . 'wp-admin/includes/file.php' );
$file = &$_FILES['my_file_upload'];
$overrides = array('test_form' => false);
$movefile = wp_handle_upload($file, $overrides);
if ($movefile) {
echo "File downloaded";
} else {
echo "when the file download error";
}
}