My PC connects to the internet through an ADSL modem. The modem itself is a gateway (TP-LINK TD-W8901G).

The gateway has a valid IP address on the web, like 66.67.68.69.
But my computer has an IP address under this gateway, like 192.168.1.13.

The gateway (or modem) is under my full control, I can open any ports or change any other advanced settings.

Is there any way to setup a webserver (for educational purpose) on my computer (being under the gateway)?

Yes. It can be done.

You'll need to use some service like dyndns.com (they give you a free address like yourname.dyndns.net), which can then be used to access your computer from the internet.

Also, if you just want to access your local computer from a domain name, then use http://lvh.me It points to your localhost. Services like these are used by web developers for ease of testing their apps with domains. So if you are running a web application at port 3500 on your computer, then using lvh.me:3500 will point to localhost:3500 :)

Hope that helps.

if you just want to set up a webserver to run scripts or tests or whatever, you don't need anything fancy for that. Just install apache (or webserver of choice) and tell your machine that http://whateveryoufeellike/ should map to 127.0.0.1 (the specific howto depends on your OS). A default apache install should also respond directly to http://127.0.0.1/ i believe. You don't need to futz around with gateways for this to work off your desktop. Of course, if you want your desktop webserver to be accessible outside your local network, that's different.

here's manual:
http://www.tp-link.com/support/download.asp?a=1&m=TD-W8901G

First check if your IP is static (i.e if ISP istn't changing it from time to time) - so you won't need to use dyndns
Then tune your modem's NAT to transfer port 80 of your computer to outer internet.
And also check if your ISP isnt blocking port 80.

To test i sometimes use hidemyass.com, to make shure my PC is available from internet.