What is the doer (fā'il)?
A verb is mentioned, and then the one who did it. That one is the فَاعِل (fā'il), and it always stands after its verb, never before it.
In جَاءَ مُسْلِمٌ (jā'a muslimun), a Muslim came, the verb is جَاءَ and the doer is مُسْلِمٌ. The ـٌ is not decoration. It is rafa, and rafa is what the role of doer demands.
The verb comes first, then its doer in rafa, then — if the verb takes one — the object in nasab.
The fā'il is not always a separate word. In جِئْتُ (ji'tu), I came, the ـتُ joined to the verb is the doer. A doer that shows as a separate word is called ظَاهِر (ẓāhir); one hidden in the verb is called مُضْمَر (muḍmar).
One thing the classical definition adds is worth pausing on: the fā'il need not do anything. It may simply be described by the verb. In مَاتَ مُحَمَّدٌ (māta muḥammadun), Muhammad died, مُحَمَّدٌ is the fā'il — yet dying is not an act he performed. Same in اِنْكَسَرَ الزُّجَاجُ (inkasara al-zujāju), the glass broke. The fā'il is the one the verb is about, whether or not it is the one acting.
One rule surprises English speakers: the verb stays singular even when the doer is plural. قَامَ الرِّجَالُ (qāma al-rijālu), the men stood — not قَامُوا الرِّجَالُ. Number is carried by the doer, not repeated on the verb standing before it.
Like most Arabic grammar terms, فَاعِل (fā'il) was taken from everyday speech and then given one precise technical sense.
In everyday Arabic it means one who does something — an agent, a worker.
فَاعِلُ خَيْرٍ (fā'ilu khayrin) — “a doer of good”
In Arabic grammar it means the ism in rafa that performs the action of the verb standing before it.