THANK YOU in sign language
Thank you, or simply thanks is an expression of gratitude. It seems that this form of respect or appreciation can be understood and expressed even by one-year-old baby or a pet. As I tore a pack of wet food, my one-year-old cat bumped her head onto my hand and stroke hard to show appreciation.
ASL signs for "thank you"
How do you sign "thank", "thank you" or thanks in ASL (American Sign Language)?
Meaning: Definition: A polite expression used when acknowledging a gift, service, or compliment, or accepting or refusing an offer; expression of gratitude or appreciation.
Pronunciation (sign description): Dominant flat hand, with its palm facing in, fingers in contact with the chin, moves forward away from the chin.
This same sign can be also expressed for thanks.
This basic ASL word THANK-YOU is also inflected to reflect the verb "THANK" in agreement with the subject-object order. E.g. THANK-HIM/HER.
Meaning: an expression of gratitude; thanks so much.
One-handed and two-handed versions with a number of movements convey a gradient of meanings in contexts. E.g. thank you so much.
If you sign THANK-YOU with both hands (and sometimes with the movement than once), it can express as "thank you very much" or "thanks so much".
If using both hands plus two movements or more, it can express as "many thanks" or THANKFUL, when one feel much appreciation. It's not a black-and-white rule, it depends on contexts and intonation.
When you express thanks or gratitude to someone for inspiration or show someone much respect for something, you TIP HAT to someone.
How do you sign if you don't want something but still show appreciation? Try "No, thanks" or "no, thank you".
Response to 'thank you'
When one says 'thank you' to you, now how do you reply back in Deaf ASL culture?
In American hearing culture, people typically say WELCOME or "you're welcome" or other forms such as "my pleasure", "no problem".
In American Deaf culture, signers usually reply in various ways as well, usually less formal (probably as Deaf community is a close-knit community that one knows each other). "Welcome" may be used but thank back is more common. Other possibilities are like "phsaw, no problem", or simply "phsaw", a quick nod with a smile, a quick thumb-up of acknowledgement, and such. They are naturally used in appropriate contexts, not something you can pick any one of them.
If you're uncertain as a hearing learner or a newcomer to Deaf community, probably the safest reply is "welcome".
Language learning, language play, etc.
Some word entries have one of some tidbits in this section, such as minimal pairs of sign words, rhymes, etc. usually related to or associated with its word entry.
[Note: ASL writing is not an official standard. This sign language writing remains in a state of open space to allow room for experiment, evolution, and improvement.]
Written ASL digit for "THANK-YOU" or "THANKS". 
Written ASL digit for "THANK-YOU".