Meaning of 'salah' and 'Salaam'

The linguistic meaning of salah is to pray and to supplicate. When it is said, salla 'alaa (صلى على) like the above ayah, it gives us three meanings:

1. To incline to someone and pay attention to them out of love.
2. To praise someone (thanaa' ثناء).
3. To pray for someone.

When salah is used for Allah ta'ala (i.e., when Allah does “salah” on someone) it means:

1. Allah loves that person.
2. Allah praises that person.
3. Allah sends blessings, His Pleasure and Mercy on that person.

When the creation does salah it means that they pray to Allah to send blessings on that creation. They seek forgiveness for them and ask Allah to send good to them. The meaning of love and praise is also understood in this meaning.Salawaat is plural of salah.

We are commanded in this ayah to do two things:

Firstly:

صَلُّوا عَلَيْهِ

“send your salah on him” this means that we ask Allah to send blessings on the Prophet sal Allahu alayhi wa sallam and ask Allah to praise him in the manner that he guided us to do.

There is a small difference of opinion amongst the scholars on the meaning of sending salah; the majority state that it means 'mercy from Allah, prayers for forgiveness offered by the angels, and duaa offered by humans.' The other group of scholars such as Ibnul Qayyim, Abul Aliyah and shaykh ibn al-Uthaymeen state that the meaning of sending blessings on the Prophet is to ask Allah to praise him in the assembly of the Angels.

Shaykh ibn al-Uthaymeen rahimahullah states,

The best that can be said concerning this is what Abu'l-'Aaliyah (may Allah have mercy on him) said: The salah (blessing) of Allah upon the Prophet (peace and blessings of Allah be upon him) is His praising him among the “higher group” (the angels). So what is meant by Allahumma salli 'alayhi (O Allah send blessings upon him) is: O Allah, praise him among the higher group, i.e., among the angels who are close to Allah.

Al-Sharh al-Mumti', 3/163, 1641

Both of these meanings can be taken as the definition because it is all included in the linguistic meaning of sendingsalah, and Allah knows best.

Secondly:

وَسَلِّمُوا تَسْلِيمًا

“and offer tasleem as it should be (offered)“. The linguistic meaning of tasleem is to offer greetings of peace. It also gives a meaning of asking for peace and security. This meaning comes from the root word of tasleem and the word salaamah. The two meanings of tasleem are:

1. 'May as-Salaam (The Perfection, One who Gives Peace and Security) be with you', meaning may His Blessings be upon you.
2. 'May you be secure, peaceful and safe.”

For the rest please see:
http://muslimmatters.org/2009/12/21/believers-send-your-blessings-on-him-a-concise-resource-for-salawat-on-the-prophet/