Is Purgatory in the Bible? 

Yes.  Although Purgatory is not explicitly called "purgatory", there are verses in Scripture which allude to it, and would create a logical fallacy if it didn't exist.  The most important reference is found in 2 Maccabees:
 

On the following day, since the task had now become urgent, Judas and his companions went to gather up the bodies of the fallen and bury them with their kindred in their ancestral tombs. But under the tunic of each of the dead they found amulets sacred to the idols of Jamnia, which the law forbids the Jews to wear. So it was clear to all that this was why these men had fallen... 
Turning to supplication, they prayed that the sinful deed might be fully blotted out... He then took up a collection among all his soldiers, amounting to two thousand silver drachmas, which he sent to Jerusalem to provide for an expiatory sacrifice ... Thus he made atonement for the dead that they might be absolved from their sin.

(2 Maccabees 12:39-43,46)

In short, if Judas Maccabee is praying and making atonement for the dead, then the proposed absence of purgatory creates the following unresolvable conundrum:

1)   A person in Hell cannot be helped by prayer
2)  A person in Heaven has no need for prayer.  

Why, then, does the Holy Spirit ask us to consider the benefit of praying for the dead?

In the New Testament, St. Paul makes the existence of Purgatory even clearer: 

...the work of each will come to light, for the Day will disclose it. It will be revealed with fire, and the fire [itself] will test the quality of each one’s work.
If the work stands that someone built upon the foundation, that person will receive a wage.  But if someone’s work is burned up, that one will suffer loss; the person will be saved, but only as through fire.

(1 Corinthians 3:13-15)

Why Is Purgatory Necessary?

Purgatory is a place of great mercy, where the soul who is not condemned, yet also not pure enough to withstand the Beatific Vision, is purged of all imperfection caused by earthly sin.  It has been said in tradition that the same fire of God "comforts the Saints", "heals the imperfect", and "punishes the damned".
 

I Was Taught That Jesus Made Atonement for My Sins on the Cross.  Why Is Purgatory Necessary?

Sin has a twofold effect:  First, it creates a debt between us and God.   Second, it has an effect which is not reversed through absolution.  Absolution cancels the debt, however, the damage to us is already done.  In the same way, Jesus dying on the Cross did not remove the ultimate effect of sin: physical death.  Whatever damage we've undertaken through our life needs to be "buffed out" in purgatory, before we can enjoy the Vision of God.  
 

What if I Don't Make It Through Purgatory?

Everyone who enters Purgatory makes it through Purgatory.  If you find yourself in Purgatory when you die, hallelujah, you're saved!  
 

Does Everyone Go to Purgatory?

No.  It is morally certain that baptized infants and children incapable of committing sin do not stop at Purgatory.  The Blessed Mother did not go to Purgatory.  It's likely (but not certain) there are many saints who went straight to heaven as well, and many un-canonized men and women who "served their Purgatory on earth" through great suffering or heroic sanctity.  The consensus, however, is that most of us who do not merit damnation still fall short of the purity necessary for the Beatific Vision.  The good news is, our souls will not resent the purging but demand it out of love for God, all the while consoled that we are saved.  

What Are Indulgences?  

Indulgences are "credits" for time spent in Purgatory which derive from the superabundant merits generated through the life of the Church, in union with the merits of Christ's Passion upon the Cross.  Indulgences are endowed on the faithful when they make certain acts of faith and devotion (usually involving an element of perseverance), usually by decree of the Pope in certain occasions or for annual feast days and the like.  In business terms, an indulgence is like a spiritual profit dividend available to the faithful, which reduces any or all  temporal punishment for our sins (depending on the indulgence).  The grace won by the saints is overflowing, and the Church generously gives us access to these merits for fulfilling a typically unburdonsome act of faith.