Philip and the Ethiopian Eunuch

Now an angel of the Lord said to Philip: "Rise and go toward the south to the road that goes down from Jerusalem to Gaza." This is a desert place. And he rose and went. And there was an Ethiopian, a eunuch, a court official of Candace, queen of the Ethiopians, who was in charge of all her treasure. He had come to Jerusalem to worship and was returning, seated in his chariot, and he was reading the
prophet Isaiah. And the Spirit said to Philip: "Go over and join this chariot." So Philip ran to him and heard him reading Isaiah the prophet and asked:
"Do you understand what you are reading?" And he said: "How can I, unless someone guides me?" And he invited Philip to come up and sit with him.
Now the passage of the Scripture that he was reading was this:

• "Like a sheep he was led to the slaughter and like a lamb before its shearer is silent, so he opens not his mouth. In his humiliation justice was denied him. Who can describe his generation? For his life is taken away from the Earth." (Acts 8:32–33 (ESV)).

And the eunuch said to Philip: "About whom, I ask you, does the prophet say this, about himself or about someone else?" Then Philip opened his mouth, and beginning with this Scripture he told him the good news about Jesus. And as they were going along the road they came to some water, and the eunuch said:
"See, here is water! What prevents me from being baptized?" And he commanded the chariot to stop, and they both went down into the water, Philip and the eunuch, and he baptized him. And when they came up out of the water, the Spirit of the Lord carried Philip away, and the eunuch saw him no more, and went on his way rejoicing. But Philip found himself at Azotus, and as he passed through he preached the gospel to all the towns until he came to Caesarea. (Acts 8:26–40).

Philip, who was an evangelist (not one of the twelve apostles), was told by the Holy Spirit to travel from Samaria via Jerusalem toward Gaza. There he was told by the Spirit to join this chariot. Philip was obedient to these directives from the Holy Spirit.
Many times the Holy Spirit gives us directions without explaining the details to us.

This man read from Isaiah 53:7b–8a:

• "Like a lamb that is led to the slaughter, and like a sheep that before its shearers is silent, so he opened not his mouth. By oppression and judgment he was taken away; and as for his generation, who considered that he was cut off out of the land of the living." (ESV).
• "Like a sheep he was led to the slaughter, and as a lamb is silent before the one shearing it, so he does not open his mouth. In his humiliation his judgement was taken away. Who will describe his generation? Because his life is being taken from the Earth." (LXX).

Whoever this is talking about, he was killed. Philip explained the meaning of these verses and Isaiah 53, which is referred to as the chapter of the suffering servant
who is Jesus Christ. It was written 700 years before the birth of Jesus Christ.
Philip probably explained the water baptism, see Christian Water Baptism 
in The Book of Romans. The man asked to be baptized, and he was. 

The Bible doesn't specify where a person must be baptized or by whom.
This means that it doesn't have to be done by an ordained person.

Philip was physically taken away by God to Azotus which is another name for the
city of Ashdod which was a capital in the Philistine region which is today's Gaza. As he went through, he preached the gospel to all the cities until he came to Caesarea
which was a port city on the Mediterranean coast. This is probably the place where he settled, at least for a while (Acts 21:8).

This is also an example where mercy triumphs over following rules and regulations, see the end of The Book of Ruth.

See also The Book of Acts and Some stories close related to the gospels 

at the end of The Four Gospels.