Here's the most basic way:
$url = "http://other-site/image.png";
$dir = "/my/local/dir/";
$rfile = fopen($url, "r");
$lfile = fopen($dir . basename($url), "w");
while(!feof($url)) fwrite($lfile, fread($rfile, 1), 1);
fclose($rfile);
fclose($lfile);
But if you're doing lots and lots of this (or your host blocks file access to remote systems), consider using CURL, which is more efficient, mildly faster and available on more shared hosts.
You can also spoof the user agent to look like a desktop rather than a bot!
$url = "http://other-site/image.png";
$dir = "/my/local/dir/";
$lfile = fopen($dir . basename($url), "w");
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, $url);
curl_setopt($ch, CURLOPT_HEADER, 0);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_USERAGENT, 'Mozilla/4.0 (compatible; MSIE 7.0; Windows NT 5.1)');
curl_setopt($ch, CURLOPT_FILE, $lfile);
fclose($lfile);
curl_close($ch);
With both instances, you might want to pass it through GD to make sure it really is an image.