What is Sharia Law? I am a Muslim myself, and I have never heard of the Sharia Law until a few people on Twitter (who were against Islam) pointed it out to me saying that it was "Islamic Law". They said that Sharia Law states to throw gays off roofs, rape women, and kill others; that being said, if that is true (which I'm sure is not), I know it is not Islam. I tried looking it up, but I only found articles by people against Islam, and I never got a valid answer, so I decided to ask a fellow Muslim. There are also no direct questions here, so the answers I found to certain questions got me confused.
The definition from Wikipedia:
Sharia... is the religious law governing the members of the Islamic faith. It is derived from the religious precepts of Islam, particularly the Quran and the Hadith. -- Wikipedia
In simple terms, there's some rules and punishments described in the Qur'an and the Sunnah, or derived from them, some of which are legally enforceable by the state---this is Sharia Law.
For example, Sharia Law for Malaysia is available online in English: http://www.esyariah.gov.my/ (click on "English" and "Reference Materials/Syariah Law"). Here's one random law from that website:
Section 23. Sexual intercourse out of wedlock.
(1) Any man who performs sexual intercourse with a woman who is not his lawful wife shall be guilty of an offence and shall on conviction be liable to a fine not exceeding five thousand ringgit or to imprisonment for a term not exceeding three years or to whipping not exceeding six strokes or to any combination thereof.
(It's the same for women.) In contrast, here's a relevant Qur'an verse:
The [unmarried] woman or [unmarried] man found guilty of sexual intercourse - lash each one of them with a hundred lashes, and do not be taken by pity for them in the religion of Allah, if you should believe in Allah and the Last Day. And let a group of the believers witness their punishment. -- Qur'an 24:2
We can see that Sharia Law, if implemented at all, is not implemented exactly as we find in the Qur'an. Nor is it enforced exactly as it is implemented. It varies from country to country, and even from region to region within individual countries.
Those specific topics you mention:
throw gays off roofs... This appears to be an ISIS innovation, along with other atrocities. (Not even the Islam-critical website thereligionofpeace.com claims otherwise.)
rape women... I think this is just made up; if anything, Sharia Law punishes zina (unlawful sexual intercourse). (Again, not even thereligionofpeace.com claim otherwise.)
kill others... Sharia Law may include the death penalty for certain crimes; it's implementation varies from place to place.
There are reasonable criticisms of about Sharia Law; this is not one of them.