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Revived as a living language in the late 19th century, Hebrew is the official language of the state of Israel, where it is the dominant spoken language (although English, Russian and Palestinian Arabic are common as well). It is also considered, along with Latin and Greek, one of the traditional classical languages of Western education. Much like them, it is often considered to have magical properties by virtue of being old and associated with an ancient religious tradition. Practitioners of Kabbalah and other bullshit like to use random Hebrew words as incantations in the mistaken belief that they actually do something. There is also an entire pseudoscience devoted to adding up random number and letter combinations in the Bible and somehow divining meaning from them, known as gematria.
Hebrew is written from right to left, which is good news for lefties but bad news for everyone else.
Lack of vowels
Before the Masoretes invented the modern system of vowel markings in the 7th-9th centuries CE, written Hebrew had no vowels. The reader was expected to just know the correct pronunciation of the word, given only its consonants were written. This is standard practice for Semitic languages, which tend to have relatively small vowel inventories.
This, coupled with its lack of use as a living language between the 1st and 19th centuries CE, is why the original pronunciation of most Hebrew words is lost to us and must be reconstructed by the comparative method, and part of the reason why the Sacred Tetragrammaton, YHWH, is pronounced "Yahweh" (the scholarly reconstruction) by some and "Jehovah" by others (based on a flawed transliteration into Latin).
Biblical Hebrew had two verbal aspects: perfect and imperfect, as well as a present participle. By the time of the Mishnah (approximately 200 CE) these had evolved into past, future, and present tenses as seen in Indo-European languages. This form of Hebrew, used by rabbis in the Aramaic- and later Arabic-speaking world, eventually was revived in the form of Modern Hebrew. Thus it had already taken on non-Semitic aspects, the same way every other Semitic language has over time: Arabic, for example, has taken on an animate/inanimate distinction. Furthermore, Modern Hebrew had to to incorporate vocabulary for things that didn't exist in the time of classical Hebrew. In some cases they are direct cognates from English, other European languages, or even Arabic. Of course, this is no different than any other languages given how influential English is in the modern world.
Nonetheless, the difference has led many observers to erroneously claim that the grammar arose from attempts by European diaspora communities to fit the grammar of their native languages onto Biblical Hebrew, and hence claim that Modern Hebrew is "not a true Semitic language" but rather a relexified Yiddish (a Germanic language). It's a plausible story, especially considering most of the early Jewish immigrants to Palestine were Ashkenazi Jews, but is historically inaccurate.
The phonology, however, is unquestionably based on the Ashkenazi Hebrew reading tradition, which shows obvious Yiddish influence in the emphatic consonants that either reduce to the same sound as their non-emphatic equivalents, or become more familiar Germanic sounds (sʼ > t͡s).
Language of babies and Adam and Eve
Because of its centrality to the Bible, a number of people have ascribed mystical powers to Hebrew. Various people including self-taught linguist Isaac Mozeson have claimed that all the world's languages originated from Hebrew. Holy Roman Emperor Frederick II attempted to raise babies without any human contact to see if they would speak Hebrew, apparently without success. James IV of Scotland tried a similar experiment by marooning children on Inchcolm Island in the Firth of Forth; one contemporary report suggested this produced Hebrew-speaking infants, although other sources say the children died.
Itamar Ben-Avi, born in 1882, was the first person to grow up with Modern Hebrew as his first language. This is largely due to the fact that his father, Eliezer Ben-Yehuda, who largely invented modern Hebrew, deliberately isolated him from any other languages growing up and spoke it exclusively at home. 
- See the Wikipedia article on Hebrew Language.
- Main page of the Hebrew-language version of Wikipedia
- Jacob Richman's list of Hebrew learning resources