It can also be used as the name of the Jewish people or "nation".
Hebrew is the language that the Old Testament was written in, although it was later translated into Greek and Latin before being rendered into the currently more-familiar English translations. The Hebrew Bible was not yet fully canonized when it was first translated into Greek, and as a result a few of the texts of the Old Testament, as found in Christian bibles, differ from the Hebrew Bible in use by Jews. The fact that it is the source language of so much of the Bible is what gives Hebrew its religious and historical importance, as many of the arguments concerning the Bible eventually refer back to the Hebrew original.
The Bible was written over hundreds of years and contains oral traditions that are even more ancient, and its language accordingly changes somewhat from part to part. Modern Hebrew is likewise not Biblical Hebrew, having amassed even more changes and lost much of its Biblical vocabulary and even some of its grammar. But these differences are relatively small, and for the most part a modern Israeli, conversant with Modern Hebrew, can pick up the Hebrew Bible and read it (with the exceptions of a few sections written in Aramaic, and an occasional confusion over words or meaning where Biblical Hebrew deviates from the Modern one). Indeed, the most substantial change to occur over all these millenia was the transition into the modern Aramaic script; ancient Hebrew was written down in a different "font", although in the same alphabet.
In comparison with English, Hebrew is noteworthy in that its alphabet contains no vowels, only consonants. While later (medieval) versions of the Bible added a system of symbols to elucidate pronunciation, the core and ancient language had none which often left words ambiguous and open to different interpretations. For example, the word הלך (he-lamed-kaf) can be read as "Halach" meaning "went" or "Helech" meaning "traveler"; such ambiguity can lead to very different interpretations of some verses. This difficulty is compounded by the fact that ancient Hebrew also didn't include marks for separating words (spaces in modern English and Hebrew), and the Hebrew Bible originally had not division into Chapters or paragraphs (such divisions were added much later).
The Hebrew Language: A Brief Overview
The Hebrew language is based around three-consonant "roots", that are put into particular patterns to convey different senses in which the core-idea is used. For example, the root "הלך" (he-lamed-kaf) signifies the idea of "walking". "Halach" (saying the first and second consonants of the root with the sound "a" as in "Cavalry") means "(he) walked" or figuratively "he went", whereas "Halcha" means "(she) went", "Holechet" means "she is walking", and so on. The various ways a word can be changed are fairly regular, so for example the root "מעד" (Mem-Ain-Daled, stumble) can be used to say "Madad" (he stumbled), "Ma'ada" (she stumbled), "Moedet" (she is stumbling), and so on. Often suffixes and prefixes are added to the root to make these changes. For example, the prefix "Ha" means "The", so "HaHelech" (ההלך) means "The Traveler".
There are a few exceptions, including exceptional words and inclinations as well as shorter or longer roots, but the core of the language lies in this way of modifying the root.
Sentence structure is flexible, but in modern Hebrew most commonly follows the form Subject-Verb-Noun. So for example just as an English speaker will say "Moses went to Egypt", a Hebrew speaker would say "Moshe [Moses] Halach [Went] LaYam [To the Sea]". Biblical Hebrew often uses a "reversed" word order of Verb-Subject-Noun, so the same sentence might be written "VaYelech [And Went] Moshe [Moses] HaYama [To The Sea, in ancient deflection]".
All nouns in Hebrew are either male or female. There is no neutral, "it", gender. So for example a table is a "he", and the sun is a "she". If you run into an Israeli saying something like "This table - she is beautiful", that is because in his native tongue (and hence in his mind) the table is a "she", not an "it".
Modern Hebrew has only three tenses - past, present, and future. Biblical Hebrew, however, had a bundle.
The Basic Sounds and Letters
Modern Hebrew has only five vowels - "a" as in "spa", "e" as in "temp", "o" as in "cone", "u" as in "cool" and "i" as in "ski". Ancient and Biblical Hebrew had a richer selection of sounds, some of which were preserved in the special symbols still used to denote pronunciation occasionally. Such nuances of pronunciation are rarely important for interpretation, however, and the five modern sounds can serve us well-enough to understand and speak Hebrew.
The Hebrew alphabet has 22 consonants/letters. Each consonant can be combined with one of the five vowels, to make for the 110 sounds the make up Hebrew words. For example, the letter Gimel (ג) can be used with the a sound to say Gamal (Camel), with the o sound to say Golem, and so on.
Actually, that's not quite right - modern Hebrew adds a few compound letters to denote foreign loan-sounds such as ג' for Giraffe, and several consonants/letters are currently read similarly though in the past their pronunciation differed. But the above "110 sounds that make up Hebrew" will do for an intuitive understanding.
As mentioned above, the vowel turning the consonant into a sound usually isn't noted down in writing. This makes understanding written Hebrew rather ambiguous at times. We will, in the following, always write down the vowels explicitly in the psedu-English notation we use, but remember that this is merely the modern or reconstructed pronunciation of the consonants.
Here then are the 22 letters of the Hebrew alphabet (in their Unicode names). The letters are written in Modern, or Aramaic, script; ancient Hebrew used the paleo-Hebrew font, a Canaaite alphabet now not currently in use that stemmed from the same Pheonician roots as the Aramaic script.
Alef (א) which we will transcribe in pseudo-English as "A" or "[ʔ]" is a continuing consonant, like "Adam". Bet (ב) actually has two forms. As written to the left it is a "soft" Bet, pronounced and transcribed like the English letter "v". If a dot appears within it, however (בּ), then it has a "Dagesh" (or, equivalently, it is Bet Dgusha), and it should be pronounced and transcribed like the letter "b". Gimel (ג) is Dalet Heh Vav Zayin Het Tet Yod Kaf Lamed Mem Nun Samekh Ayin Pe Tsadi Qof Resh Shin Tav
Some Basic Roots, Words, and Structures
Let us begin with some of the most common roots in the Hebrew Bible. A few verbs:
Alef-Heh-Bet (אהב) - meaning love.
And a few nouns:
Aleph-Lamed (אל) - god, but also towards and with the a vowel (al) means "don't".
See 500 Basic Words for more example of highly-relevant words.
Let us now consider how to alter them according to a few common structures.