Bengali is the official language of Bangladesh, the state of West Bengal in India, and India itself (though it has others as well). There are also signifigant numbers of Bengali-speaking people in the bordering Indian state of Assam and in Bengali immigrant populations in the Middle East and West. There are a total of 190 million native Bengali speakers, making it the 5th most common language in the world. "Bengali" is an english word that means a Bengali speaker and the name of the language itself, which is called "Bangla" in its own language. (Handy tip: It's not pronounced Bang-la, but sounds closer to bahng-la)

Bengali is an Indo-European language. It is usually written in the Bengali script. This is a Brahmic script, very similar to Devanagari, which is used for Hindi and Sanskrit. Each base symbol represents a syllable, and other symbols can be added to change (or suppress) the vowel of that syllable. Consonant clusters are often indicated by joining two symbols. The spelling leaves out some vowel merges in the spoken language, so there is not a true one-to-one relationship between the written and spoken word.

There is a difference in dialects, there is a sort of colloquial form and a literary, often formal, form. While the colloqial form is overwhelmingly from Sanskrit, there are also plenty of words taken from English, Hindi, Arabic, and Persian sources. Bengali also has many regional variations, particualry between Bangladesh and West Bengal, and I hear Bengali spoken in the UK has more Arab-Persian words in it and is different from "regular" Bengali grammar. The main difference is the sound, though Muslim Bangladesh may use some different words than Hindu West Bengal, some of it from Urdu. Bengali has a history related to Urdu; when Pakistan and Bangladesh were united, the central government tried to establish Urdu as the official language, leading to large protests. Bangladesh and West Bengal even commemorates February 21 as Language Martyr's Day, to remember those killed while protesting to make Bengali the official language.

Some useful Phrases:

Hello: Assalaamu 'Alaykum (typical Islamic greeting)
Hello: No Mosh Kar (secular)
Nice to meet you: Apnake dekhe kushi holam
How are you?: Kemon Acho? (informal) Kemon Achen? (formal)
Yes: Han
No: Naa
My name is ...: Amar nam ...
What’s your name?: Apnar nam ki?
Are you crazy?: Tumi ki pagol? (informal)
Oh, how beautiful!: Ah, ki shundor
I don't know Bengali: Ami bangla jaani naa (remember, bahn-gla not bang-la)

Useful links and sources: