Sign language alphabet also known as manual alphabet is used to fingerspell a series of the alphabetical letters to form a spoken/written word of another language. Fingerspelling is used when an Ameslan (Deaf/ASL-speaker) refers to an English or a foreign word. For example, a person's name, a place's name, a word of another spoken/written language, etc.
Just as you speak English, you'd occasionally come to a lack of English word for a foreign city, country, or place, a person's foreign name, or a concept that doesn't exist in English but exists in another language, you'd insert the foreign word into your English utterance. That goes the same for ASL that uses fingerspelling for a foreign written/spoken word.
There are different sign language alphabets around the world. Some countries or sign language cultures use similar manual alphabets with a few modifications, borrowing the alphabet from one another. Some are entirely different.
Some manual alphabet systems are one-handed. Some others are two-handed. One-handed sign language alphabets are used in the U.S., Canada, and many other European countries. The one-handed American manual alpabet is used in Canada and the U.S.
The two-handed British manual alphabet is used in Australia, England, New Zealand and some other countries. The British manual alphabet and British sign language (BSL) are entirely distinct from the American manual alphabet and American Sign Language (ASL) which are used in North America (Canada and the U.S.).
Cross-cultural fingerspelling: It can take one just a few moments to learn a manual alphabet, but it takes one a bit longer to learn to "read" the fingerspelled word of another system.
At the World Games for the Deaf in Christchurch, New Zealand in January 1989, some Australians and New Zealanders had learned American manual alphabet beforehand while some Americans and Canadians learned British fingerspelling. Why?
At a social gathering, I chatted with an Australian in mixed gestures, onomatopoeic signs and others in combination, because neither of us know each other's languages (Ameslan/ASL and Auslan). Whenever he could not understand what I said, I fingerspelled an English word in British alphabet. He read and understood it naturally. Likewise, when I could not understand him, he fingerspelled in American alphabet that I could understand easily. After all, we read and write the same language, that is English.
But, you might scratch head, wondering that since we know the same language English, why do we use each other's manual alpabets? Unlike the whole words, the fingerspelled words are letter-by-letter based. Like hearing learners, we are not accustomed to read the fingerspelled words that fast from another manual alphabet. We need to practice reading letter by letter and develop the receptive skill in another alphabet. Unlike fingerspelling, fluent signers could understood each other in signs/words at a regular pace. Interesting, eh?
"Which right or left hand should left-handed people use?"
Left-handed individuals remain to be left-handed in fingerspelling and sign language, while few other left-handed people are right-handed signers. If you are a left-handed, your left hand would be the dominance in fingerspelling and sign language. Likewise, if you are a right-handed, use your right hand as the dominant one.
Related posts: What is the difference between fingerspelling and Fingerspelled loan?