Conversion in Islam is very easy and simple. All it requires is stating the testification of faith (الشهادة, ash-shahaadah), which happens to be the first Pillar of Islam (ركن الإسلام, ruknu-l-islaam; plural: اركان الإسلام, arkaanu-l-islaam).
This testification is made of two parts. The first part testifies that there is no god but God/Allah. The second part testifies that Muhammad (Muhammad b. ‘Abdillaah, the founder-prophet of Islam) is the messenger of God.
(Random note: most South Asian translations translate the first part as saying that nothing worthy of worship other than God, and the usual translation of the second part calls Muhammad God’s apostle. I don’t know why South Asians prefer “apostle” to “messenger”; I personally find “messenger” to be more accurate and easier to understand.)
The testification must be said in Arabic. Furthermore, this is not the shahaadah one finds ubiquitously in the Islamic world: it’s a special version. In the usual shahaadah, the first part says: لا إله إل الله, laa ilaaha illa’llaah; in the testification, it begins with اشهد أن, ash-hadu an (“I testify that…”). In the usual shahaadah, the second part says simply: محمد رسول الله, muhammadu-rasooli-llaah; in the testification, it begins with وأشهد أن, wa ash-hadu an (“and I testify that”) and continues with: محمد عبده ورسوله, muhammadan ‘abduhu wa rasooluhu (“Muhammad is his slave and messenger”).
With this, in front of witnesses, the conversion is done. There are some suggestions which are often followed. Many converts take a Muslim name. Some legally change their name, others don’t. Some don’t change it for propaganda purposes: they look and sound like non-Muslims, and have a non-Muslim name, thereby winning non-Muslims’ trust, and then preach about Islam to win converts. (I heard this during a conference aimed at teaching about propagating Islam.) Although, unless one’s name is obviously anti-Islamic (Christopher, Ramdas, etc.), the change in name is not required. One is also obligated to take a bath (غسل, ghusl) which with the shahaadah purifies one of all sins.
I’ve been to a few conversions to Islam. While the time is takes to take the shahaadah is about the same it takes to be baptized into Christianity, the attendant ceremonies and observances make conversion to Islam somewhat underwhelming. I’d suggest Muslims take a look at conversion in Christianity or Judaism and modify it accordingly. Not that the actual rite has to change, but the attendant observances should change a bit to make it more glamorous and inspiring. A lecture (which more often than not revolves around the lecturer’s accomplishments in life and number of converts and less on the convert-to-be) and coaching the convert-to-be in saying the shahaadah does not give the rite the dignity it deserves. But that’s just my opinion.