What is "mysqli"
Mysqli is one of the option to connect to mysql besides PDO, the mysql_* functions have already been deprecated and soon will be removed it is highly recommended to migrate the code to use a newer API.
There are two styles for working like Mysqli the object-oriented and procedural.
Procedural example
$conexao = mysqli_connect('localhost', 'usuario', 'senha', 'banco');
Example OO
$conexao = new mysqli('localhost', 'usuario', 'senha', 'banco');
When migrating a code with mysql_* functions to mysqli_*, be aware that when using the procedural style most functions are first argument is always the connection.
Try using Prepared statements to avoid problems with sql Injection.
//código legado
$result = mysql_query('SELECT * FROM produtos') or die(mysql_error());
//Exemplo com mysqli
$result = mysqli_query($conexao, 'SELECT * FROM produtos') or die(mysqli_error($conexao));
Recommended Reading
Manual - Mysqli function/resource list