Can you put this Father, Son, Holy Spirit thing in terms I can understand?

The Father is the Creator of the Universe and has taken an interest in mankind since the beginning. He is Good, powerful and wise, capable of absolutely anything.

The Son was born to a woman named Mary who was a virgin. Since this was in the days preceeding artificial insemination, this was considered miraculous and only could be explained by the direct intervention of God. The Son grew and was called Jesus, who lived an ordinary life as a carpenter -- although he was a very bright child. In His early thirties after his cousin John the Baptizer began teaching, Jesus went to hear John and was baptized (immersed in water) by John. At this time Jesus began teaching and continued this for three years.

After Jesus rose from the dead (known as the Resurrection), He gave His Disciples (His remaining 11 closest students) and several hundred other followers the Holy Spirit into their hearts to guide them. The Holy Spirit is still present in believers.

The concept of three parts to God is called the Doctrine of the Trinity.

Are there any good modern day analogies to this Doctrine of the Trinity?

Yes. Perhaps the best one for computer-savvy readers is this:

In the Beginning, a Program existed. It consisted of three parts: a Server, an Installer, and a Client. The Server was originally on the Net. It communicated with its operator, but the operator sent the Server illegal requests which violated its Protocol. The operator was locked out by the Server.

Over time, various operators tried to communicate properly with the Server, but to no avail. Finally, the Server choose an operator to whom He published a file full of a communication Protocol for various operators and programmers to use to communicate to the Server. This key operator even published a detailed user's manual explaining the Protocol. But the chosen programmers would not stick to the Protocol or read the manual, and kept trying other protocols and sending illegal requests to the Server. The Server would not allow access because of this, and responded by crashing their systems.

Finally, the Server sent an Installer program. The Installer was sent to the programmers and operators to show them the proper way to communicate with the Server and never have system crashes, but by now the programmers and operators wouldn't listen. So they deleted the Installer, assuming that it was a malicious virus.

Shortly thereafter, the Installer re-appeared, un-deleted by the Server. The Installer went to several key programmers and operators who had listened to it and proceeded to Install the Client on their systems.

The Client remains and has become widely used throughout the world. People who use the Client and study the manual experience trouble-free communication with the Server. These people also know that the Installer will return to the Net some day with a new Version which will usher in a new and wonderful age.
The Server is the Father.
The Installer is the Son -- Jesus Christ.
The Client is the Holy Spirit.
The manual is the Bible.

All are necessary. The Father is Supreme. Belief in The Son is necessary to establish proper communications with the Father -- belief shows proper respect. The Holy Spirit is within us for guidance in life and in those communications. And the Bible is the instruction book.