What is Jihad in Islam

First of all we have to differentiate between a belief or a doctrine or an idea and its followers. You do not judge the validity or worth of an idea based only on the manners of one or group of its' followers, while you can judge a follower by how honest he is to his belief.

Why is that? because any one can claim to believe in something or embrace a certain religion just for benefits that he expects to have, in this case he is a hypocrite and a liar. Sooner or later such characters will be exposed because their hunger for benefit will sure manifest itself one way or the other.

Also someone can really believe in a certain idea but, being human, he may not be able to stick to his belief 100% of the time, at some points he can deviate because nobody is perfect. You cannot fairly judge that idea or belief by judging the follower at a point when he was vulnerable.

For example, all religions forbid adultery, can I fairly say that all religions are wrong because there are people who commit adultery while being for example Jews or Christians.

You can judge an idea, belief or religion by how far it changes people, make an impact on their lives. God does not want us to be perfect but He wants us to stick to the pattern of conduct that guarantees a peaceful co-existence to everybody. He wants us to admit when you make a mistake and to accept the consequences of your deed and believe that if we honestly regret our deed then He will forgive us that's why He called himself the All-Forgiving.

Prophet Muhammad, peace and blessings be upon him, said: "I swear by Him in whose hand is my soul, if you were a people who did not commit sin, Allah would take you away and replace you with a people who would sin and then seek Allah’s forgiveness so He could forgive them."[Sahih Muslim (2687)]

The definition
The word Jihad comes from the root j / h / d, which in Arabic means to exert the most effort or struggling or striving. This definition is general as one can exert effort in studying or fulfilling goals and ambitions in a variety of areas.

The Arabic word for war is: "al-harb" or "al-qitaal" NOT "Jihad"

In Islam the idea of exerting effort has two levels, a major level and a minor one. The major level of Jihad is jihad al-nafs or struggling against one's lower self and its demeaning lustful desires. This Jihad is the hardest because it needs discipline and hard work. The lesser, or minor Jihad, is al-qitaal or armed struggle. This is the Jihad that has been attacked by unjust and misleading propaganda in an effort to equate it with mere bloodshed.

The General Rule
Not all wars are considered "Jihad" unless it follows Jihad's rules, boundaries and conditions (which will be discussed below). If these conditions and rules are not followed closely, then this is NOT considered "Jihad", actually in this case it will be considered as corruption and treason that deserve to be punished.