Musalla refers to a temporary place of prayer (salah).
The prayer rooms (musallas) are set up in offices and schools, etc and they do not come under the rulings on mosques because they are not established as waqfs (Endowments) and as such they do not cease to be the property of their owners. Moreover, the five daily prayers are not held regularly in them in most cases.
And take, [O believers], from the standing place of Abraham a place of prayer. Surah Al Baqarah 2:125
The entire earth is considered a masjid in the general meaning of the word. However the specific meaning for masjid is a place purposely built for prayers to be held at all times. A masjid is a place specifically built for prayer. It can be built by stone, mud, or cement. Even if the place is designated to be a masjid and it isn’t complete, but intended to be a masjid then it’s a masjid.
The musalla on the other hand is a place that a person conducts his prayers at sometimes. This place isn’t considered a masjid. He only prays there when the prayer comes in (and he can’t make it to the masjid). The proof for a musalla can be found in the Prophet’s action(peace and blessings be upon him). He used to pray his supererogatory in his house, and his house wasn’t a masjid. Also Utban ibn Malik (may Allah be pleased with him) invited the Prophet (peace and blessings be upon him) to his home , so he could pray in the place Utban used to pray. That place at Utban’s home wasn’t a masjid.
Therefore a musalla is a place where people conduct their prayers, and this place isn’t designed or designated to be a masjid. Additionally, the people who pray there know it’s a musalla.