In the Christian Bible, the first Pentecost is marked by the incident of "speaking in tongues" (glossolalia), as described here in the Book of Acts:
When the day of Pentecost came, they were all together in one place. Suddenly a sound like the blowing of a violent wind came from heaven and filled the whole house where they were sitting. They saw what seemed to be tongues of fire that separated and came to rest on each of them. All of them were filled with the Holy Spirit and began to speak in other tongues[a] as the Spirit enabled them. Now there were staying in Jerusalem God-fearing Jews from every nation under heaven. When they heard this sound, a crowd came together in bewilderment, because each one heard them speaking in his own language. Utterly amazed, they asked: "Are not all these men who are speaking Galileans? Then how is it that each of us hears them in his own native language? (read more...)Peter goes on to explain to the crowd that this is a sign of the Holy Spirit being poured out upon them, and that anyone who is baptized in the name of Jesus Christ can receive the gift of the Holy Spirit.
Pentecost is observed with local variations in the Catholic and Orthodox churches; you can read about these different observances in the wikipedia article.
The name "Pentecostal" has also been adopted by an evangelical Christian movement which is focused on direct personal experience of God as manifested by the baptism and gifts of the Holy Spirit. (Some Pentecostals baptize in the name of Jesus only, while others baptize in the name of the Father, Son and the Holy Spirit.) Pentecostal prayer and worship service tends to be highly emotional, demonstrative and enthusiastic. The Pentecostal movement emerged from the revivalist "Holiness movement" in Protestant Christianity, and began to take its distinctive shape in the late 19th century. In 1906, there was a front-page article in the Los Angeles Times which reads as follows: "Weird Babel of Tongues, New Sect of fanatics is breaking loose, Wild scene last night on Azusa Street, gurgle of wordless talk by a sister." You can read more about the teachings and history of Pentecostalism at the wikipedia article. Although it is not one of the historical Protestant churches (Baptist, Methodist, Lutheran, etc.), it has a very large membership and is continuing to expand as a world-wide Christian movement.
As for English "WhitSunday," the name is short for "White Sunday" and seems to refer to the fact that those who were baptized at the festival of Pentecost were dressed in all-white robes. The first Sunday after Easter is called "Dominica in albis" ("Sunday in whites") for the same reason. Whitsuntide is a related word that refers to the festival of Whit Sunday plus the days that follow, also known as Whitsun Week.
For an image, here is an image from an illuminated manuscript from the late 15th century showing the first Christian Pentecost: