the mote in thy brother's eye
Matthew 7:3 (similar statements in Matthew 7:4, Matthew 7:5. Luke 6:41, and Luke 6:42):
And why beholdest thou the mote that is in thy brother's eye, but considerest not the beam that is in thine own eye?
This is about not using arbitrary criteria for putting other people down. We are all imperfect, and before we judge other people we should apply those same standards to ourselves, and not require other people to meet standards that we can't meet.
This is a key concept in social justice: the universal applicability of social standards and rules, rather than one set of rules for me and another for thee.
casting the first stone
So when they continued asking him, he lifted up himself, and said unto them, He that is without sin among you, let him first cast a stone at her.
The Pharisees had brought in a woman accused of adultery, and asked Jesus what to do with her -- apparently trying to get him to defend her, which would have put him at odds with the authorities.
Instead, he basically makes exactly the same point made in Matthew 7:3: before you go condemning others for their sins, apply the same standards to yourselves.
the least of my brethren
Then shall the King say unto them on his right hand, Come, ye blessed of my Father, inherit the kingdom prepared for you from the foundation of the world:
For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in:
Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me.
Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee? or thirsty, and gave thee drink?
When saw we thee a stranger, and took thee in? or naked, and clothed thee?
Or when saw we thee sick, or in prison, and came unto thee?
And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me.
Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels:
For I was an hungred, and ye gave me no meat: I was thirsty, and ye gave me no drink:
I was a stranger, and ye took me not in: naked, and ye clothed me not: sick, and in prison, and ye visited me not.
Then shall they also answer him, saying, Lord, when saw we thee an hungred, or athirst, or a stranger, or naked, or sick, or in prison, and did not minister unto thee?
Then shall he answer them, saying, Verily I say unto you, Inasmuch as ye did it not to one of the least of these, ye did it not to me.
In summary, Jesus (referred to here as the "son of Man", verse 31) is saying that all people are to be treated as you would treat the highest nobility or even Jesus himself. If you don't feed and clothe the poor, then it's like turning away Jesus himself.
The secular translation of this: take care of everyone.