Here are some approaches to recording electric guitar on a computer:
- Micing an amplifier and recording the sound from the mic
- Using a multi-effects pedal with USB audio connection like Boss GT-100, Line6 POD HD, or Zoom G3
- Plugging the guitar into a soundcard that has a high impedance input and using amp modelling software
Micing an amp is a traditional recording technique but requires skill. You need a sound card, microphone, and amplifier that sounds good. High chance of noise or poor sound unless you learn your gear plus spend time and money.
Multi-effects pedals are the simplest way to do it. Prices are probably between 100-200 EUR for the low end gear, maybe even cheaper if you take a 3.5mm line out from a cheap multi-effects pedal.
Connecting the electric guitar to the soundcard and using amp simulation software on the computer gives you a lot of choice. Usually a built-in sound card will not perform well because it is noisy and doesn't have a high impedance input that mimics a guitar amp input. GuitarRig was mentioned, S-gear is my favorite software amp at the moment. There are free software amp VST plugins like LePou's Hybrit.
There are many factors involved, you will need to research the options yourself by comparing products and trying them out where possible.