Assalamu Alaikum - The Greeting of Muslims!
This is the greeting that Allah Ta'aala has blessed Muslims with. It was initiated from the time of Adam Alayhis Salaam and will continue in Jannat. This is also the greeting that the angels use.
One should try to understand the excellent virtues and merits of 'Assalaamu Alaikum'. When a person greets another with "Assalaamu Alaikum," it is not simply a token of love, rather it is a reminder of Allah Subhaanahu wa Ta'aala.
How? The word As-Salaam is actually the name of Allah Ta'aala given in the Qur'aan. When you are greeting another Muslim, you are also remembering Allah Ta'aala. In a gathering when a person remembers Allah, he is the best person. So in effect, this is why the person who greets first will be the better of the two people who are greeting. This is not only a rememberance of Allah by the one greeting, but it is also a reminder of Allah Ta'aala to the one he is greeting.
'Assalaamu Alaikum' is not only a greeting; in fact it is also a du'aa. 'Assalaamu Alaikum' means 'May peace be upon you'. This means you are making a du'aa that this person is taken care of and protected by Allah.
One may also add the words 'Wa Rahmatullahi wa Barakaatuh' to 'Assalaamu Alaikum'. This would be extending the du'aa; along with Allah's protection, you are asking Allah Ta'aala to send His mercy and blessing upon the one you are greeting.
After having understood all that is implied when greeting someone with 'Assalaamu Alaikum' or 'Assalaamu Alaikum wa Rahmatullahi wa Barakaatuh' it is impossible for the one greeting to have enemity for the one he is greeting.
May Allah Ta'aala give us tawfeeq to follow upon the beautiful sunnah of greeting fellow Muslims with 'Assalaamu Alaikum'. Ameen.
*Make Dua, Do your best ,And leave the rest to ALLAH!