American Sign Language, or ASL, is the language of deaf communities across most of North America. It is a descendant of French Sign Language, mixed with the local sign languages, officially born in the early 19th century with the founding of the American School for the Deaf. As with all sign languages it is a true language, not a code for or even related to spoken American English.
Sign languages, like spoken languages, have phonology even though they obviously contain no noises. In the past linguists have used the terms cherology and cheremes (in place of phonology and phoneme, from Greek for "hand"), but these terms have fallen out of use and ultimately obscure the fascinating relationship between the audible building blocks of spoken languages and the spatio-temporal ones of sign languages. So phonology and phonemes it is. Like with spoken phonemes there are also sign allophones, predictable variations in signs that do not effect meaning. While when describing phonemes we use terms for the parts of the mouth used to make the sound, the level of airflow obstruction, and the vibrating (or lack thereof) of the vocal chords, signs are built out of handshapes, hand orientations, the location the sign occurs in relative to the signer, the movement the sign goes through, and even non-manual motions, things like facial expression or the orientation of the body. That final category is mostly used for things that in spoken language would be conveyed through stress or intonation.
A clever person, armed with the knowledge that sign languages have phonology just like, or at least analogous to, spoken languages may wonder if they could then be written. After all I did say it's not like English, and written English is very closely related to spoken English, and thus is also unlike ASL. The simple answer is yes, yes it can. The most common one is called SignWriting, or Sutton script, developed by a dancer called Valerie Sutton who adapted it from her notation used to transcribe dance movements. SignWriting uses a large number of iconographs with notation for space and movement. Individual signs are arranged in two dimensions, then can be arranged in linear order. The difficulty with SignWriting is it has a very large inventory of symbols, making learning it similar to learning Chinese characters, and the two dimensional signs require special software, limiting accessibility. That said, Unicode does support it (as an auxiliary). There are other systems, my favorit of which is Stokoe notation, which can be written much more easily than sign writing and is almost as comprehensive. It is also a good excuse for me to throw a bone to William Stokoe, the 20th century linguist who did for Sign Languages what William Labov did for African American Vernacular; gave it a voice in academia and forced, through research, the acceptance of its legitimacy.
ASL has Subject Verb Object (SVO) word order, though some would say that Topic-Comment word order is more accurate (I think you can figure out what that means without more diatribe on my part). ASL lacks any grammatical tense but has many ways to conjugate for "aspect" i.e. the flow of time. So while without explicit mentioning you wouldn't know if the action of the verb took place in the past, present, or future, you will know if I've just begun to do it, if I do it habitually or as a single instance, if its an incomplete action, and so on. Most of these are done by manipulating the speed and number of repetitions of a sign. Pronouns are largely made by pointing to the present parties; if the referent is not present the signer will designate a place or "locus" and point to that whenever they need to refer to that individual for the remainder of the conversation. ASL has a system of "classifier" handshapes, an inventory of handshapes that represent categorical things (objects, vehicles, animals, people, etc.) that can be mixed with other signs productively.