Say what?
Here are some common words and phrases that Muslims use in everyday language. Non Muslims usually don't know the meanings of these words, so I thought I would post them here so you know what we are talking about. :)

Alhamdulillah- Literal translation: Thanks be to God. Pretty much, we say it in place of 'thank God'. It shows our thankfullness and appreciation to God.
(ex. Alhamdulillah I passed my test!)

Inshaallah- Literal translation: If Allah (God) wills. Or, in english we usually say it as: God willing.
(ex. I am going to the store later, inshaallah)

Mashaallah-Literal meaning is: God willed it. This is usally used when we are praising someone or something, and reminds us that everything is from Allah.
(ex. You look so nice today, mashaallah)

Subhanallah-Literal meaning is: Glory to God. It is also a form of praising.
(ex. SubhanAllah, look how many stars are out tonight)

Jezak Allah Khair-Literal meaning is: May Allah grant you goodness. It's the best way to say 'thank you' to another Muslim.

As salaamu Alaikum- The greeting that is used to greet another Muslim. Means, peace be upon you.
Wa laikum as salaam- The response to As salaamu alaikum, which means, 'and upon you be peace'.

Hadith- A report of the sayings or actions of Muhammad (peace be upon him) or his companions, together with the tradition of its chain of transmission.
The classification and authentication of the hadith is then of crucial importance to the shariʿa. As most reports were collected about 150 years after the death of the Prophet, a number of disciplines collectively known as the sciences of the hadith were developed, specializing in external criticism (investigation of the isnad, biographical studies of the reporters and of their characters, historical context of each report and each subsequent transmission) and internal criticism (consistency with the Qurʾan, consistency with other hadith, historical consistency). Depending on the findings of these various studies, a hadith would be classified as sahih (authentic), hasan (good), daʿif (weak), and mawdu or batil (forged). Six main collections of the hadith gained wide acceptance, and of these the Sahih of Muhammad ibn Ismaʿil al-Bukhari (d. 869) and the Sahih of Abu alHusayn Muslim (d. 875) are the most authoritative.

Why do Muslims say these things?

Saying these words and phrases are actually a form of worship and glorification of God. When we say, 'oh wow, look how pretty the sunset looks, mashaallah!' it is giving praise to God who made the sunset look that beautiful. We don't have to say it, but it's better to, because it reminds us of God and also praises him (worship).

Why do we always say 'peace be upon him' after mentioning any of the prophets?

We say this as a form of respect for the prophet that was mentioned, and to remember that while they were all human like us, they were special, and we pray that God gives them peace and blessings.