If it's just a CSV file on a website, cURL might be a little overkill (assuming allow_url_fopen is On)...
$csv = file_get_contents('http://thesite.com/path/to/file.csv');
Previous suggestions for importing CSV's into a SQL database would be to use MySQL's "LOAD DATA INFILE" syntax to load data from a .csv file.
To do this, you'd obviously have to write the contents of the CSV to a file on the server before running the "INFILE" SQL query. Otherwise, you might look at using file() instead and simply iterating through the .csv, INSERT'ing rows as you go.