Verily, verily, I say unto you, He that entereth not by the door into the sheepfold, but climbeth up some other way, the same is a thief and a robber.

The sheepfold is the collection of the saints.

and Jesus is the door?

Yes  (10:7)

According to ESV Study Bible, the literal definition is that it was a courtyard next to a house, where the family's sheep are kept, to protect them from predators, etc. Sometimes caves or other natural formations were used. The gate would be guarded by someone, sometimes the shepherd.

I also heard a sermon, I think about Psalm 23, where the pastor described how the shepherd would stand next to the gate and look each sheep over for wounds and anything needing attention as the flock was brought in for the night. IIRC, that's what "anoints my head with oil" comes from, although it might be referring to purification in the passage. Olive oil was used as a wound dressing.

It basically means do not follow a false teacher or a false gospel.

We, the chosen, are His sheep.  He will enter through the gate and we will know His voice.

He is the only one that can enter through the gate.  Anyone that enters by means of climbing over
a wall is a false teacher; a false prophet; one who teaches a false gospel.

Only follow Christ.