Making two congregations in a mosque is disliked by most scholars. Some of them might allow it if the congregations were not at the same time others would categorically consider it as disliked.
So basically if you are in a mosque and a congregation prayer is done you are asked to join it else if you missed the congregation with the regular imam at the mosque you should pray for yourself. Except for the case you are praying at home or at a room were people can pray or a mosque without a regular imam!
Now some madhabs or scholars these days allow to join a congregation even if you don't have the intention to pray the same kind of prayer. For example often in Ramadan people who missed the actual 'Isha' prayer with the Imam join him in the Taraweeh prayer with the intention to pray 'Isha' and offer two more raka'a.
Others consider this as invalid as the Imam is intending to pray a nafl or sunnah (recommended) prayer while you are praying a fard (ordered) prayer. Their solution is to pray 'Isha' alone and than follow the Imam.
Note that some madhabs and scholars even say you can't perform two congregations for the same prayer (time) in the same mosque. That means even if you entered the mosque and the prayer was finished and the congregation left and you met a couple of people wanting to pray together they would say no each of you must pray alone, as the "congregation" has finished.
The issue of the second congregation in a mosque
The difference is based on one important thing that this mosque has a paid or regular Imam and a regular Muadhin!
Their point is: The prophet () always lead prayers in his life and whenever sahaba came late and missed the congregation with him they prayed the prayers alone. It was even narrated that many of them came and each one prayed the missed prayer for himself!
Imam a-Shafi'i whom's madahab is one of those allowing the opposite view (to some extent as you may read) even pointed at it in his book al-Umm الأم saying:
وإذا دخل جماعة المسجد، فوجدوا الإمام قد صلى صلوا فرادى، فإن صلوا جماعة أجزأتهم صلاتهم، ولكني أكره لهم ذلك، لأنه لم يكن من أحوال السلف
( - My own translation take it carefully - )
if a congregation of people entered a mosque and found that the Imam has already ended the prayer, they pray each of the alone, but if they prayed as a congregation their prayer is valid, but I consider it as makruh as the salaf never did the same!
وأما مسجد في قارعة الطريق-ليس له إمام راتب ولا مؤذن راتب-فلا بأس من تعدد الجماعة فيه
But if it was a mosque at the side of a road without a paid or regular Imam and mua'dhin then it would be fine to perform multiple congregations in it.
إنا قد حفظنا أن جماعة من اصحاب النبي صلى الله عليه وسلم فاتتهم صلاة مع الجماعة، فصلوا فرادى مع أنهم كانوا قادرين على ان يجمعوا فيه مرة أخرى، لكنهم لم يفعلوا ؛ لأنهم كرهوا أن يجمعوا في مسجد مرتين
And we guard (memorized) that a congregation of the companions of the prophet () have missed a prayer in congregation, and then prayed each one alone even if they would have been able to pray together in a congregation, but they didn't because they disliked performing a second congregation prayer in a mosque!
If we consider the importance of the congregation prayer (read for example this hadith) we could also conclude that doing or allowing a second congregation would have a negative impact on the first congregation, as people would come late because an other congregation would be available!
Therefore this view (the 2nd congregation being makrooh) represents the view of the majority of scholars among them abu Hanifa, Malik, a-Shafi'i and imam Ahmad as reported by his student abu Dawod a-Sijistani from him.
While the opposite view is based on this hadith, which is seemingly allowing an other congregation and this is the known opinion of Imam Ahmad which was reported by most of his students. Note that the person whom joined the man in that hadith had already prayed with the congregation as the hadith seem to indicate, so for him the prayer could be a sunnah/nafl! So the case imam praying fard with ma'mum or congregation praying fard isn't covered by this hadith at all! As you see this even would reject what people do in taraweeh by following an Imam who is praying sunnah/nafl while they are praying fard!