How is the Tahiyatul Masjid salah performed and what is the additional rakka some people perform?

Question

I am new to the Muslim world and I am still learning how to perform Salah correctly. So I have a few questions:

1. When entering the mosque, I should do Tahiyatul Masjid. 2 rakkats, finished with "assalamu aleikum wa rahmatullah". Correct?
2. I should do as many rakkats as needed in salah behind imam. This is clear.
3. I saw some people after the salah still praying again, I think one rakka. What is that? Is that the regular "rakka text"?

Answer 1

Well you should know that in Islam we have five fard or obligatory prayers which are well defined (amount of raka'a etc.).

After that we have a couple of prayers which are highly recommended.

These prayers are related to the timings of the five fard prayer and are called sunan rawatib (here scholars may have different opinions on the amount which counts as such and how to: set of two raka'a/set of four etc.). Some of these fard prayers have sunan rawatib which should be performed before the fard prayer, some have sunan rawatib after the fard prayer and some have both. See What are Sunnah Rawaatib?. None of these sunnan rawatib is only one rak'a, unless you meant witr, but usually this is prayed at least after a set of two rak'a.

So maybe what you have observed are people who came late and caught up the prayer one or more raka'a later (See What is the exact manner to join the Salat Jamat after arriving late?), as one is recommended to keep in a sitting position to do some dhikr after finishing prayer before moving to sunnah, so usually only those who joined the prayer after it has started would move up once the imam has performed the tasleem(s) end ended the congregation prayer. As anybody is asked the pray the prayer as a whole so if he missed one or more raka'a (or was behind the Imam with this amount) he should complete his prayer until he prayed the full fard amount!

The tahiyyat al-Masjid is also a highly recommended nafl, it must be done each time you entered a Mosque if you have enough time to do it before the congregation prayer starts. Scholars say if you entered the mosque several times it is recommend to do it each time, but for example if you need to cross a mosque many times a day you may do it at least once. It is a set of two raka'a which you perform silently.

Maybe helpful:

What are the rules for reading 4 rak'a fard with jamat or alone?

Catching Salat just before the last word of last Salam

Do we have to say 'Allahu Akbar' loudly in every rakat of every salat?

Why do people pray salah differently?

How can we offer Fard Namaz with just two people

Is it permissible to split a 4-rakat farz salat into two 2-rakat farz salat?