Since you've tagged your question 'php', I'll assume your running php on your server. Your best bet is if you control your own web server, then compile cURL into php. This will allow your web server to make requests to other web servers. This can be quite dangerous from a security point of view, so most basic web hosting providers won't have this option enabled.
Here's the php man page on using cURL. In the comments you can find an example which downloads and image file.
If you don't want to use libcurl, you could code something up using fsockopen. This is built into php (but may be disabled on your host), and can directly read and write to sockets. See Examples on the fsockopen man page.