Aitäh: This is one of the more common ways to say "thank you" in Estonian. If you are just looking for a quick, simple phrase to memorize for "thank you," this is a good one to use. Originally a short form of aitüma or aituma (see below), it is now more common than either of those. Its pronunciation is unique for Estonian because normally the letter ä comes only in the first syllable, and the stress of Estonian words is usually on the first syllable as well; in the word aitäh both the stress and the letter ä come in the second syllable: aitäh.
Suur tänu: This means "big thanks," and is commonly used as a way to say "thank you very much". Similarly, you can also say Suur aitäh. To show even more gratitude you can say Suur-suur tänu or Suur-suur aitäh. There is also Palju tänu, "many thanks," Tuhat tänu, "a thousand thanks," and Südamlik tänu, "heartfelt thanks." Note that the correct word for all of these expressions is tänu, not tänud.
Tänan: This comes from the verb tänama, "to thank," so it means "I thank you." You can use it on its own or you can say Tänan väga "thank you very much", Tänan teid "thank you" (formal or plural), or Tänan sind "thank you" (informal). Ma tänan teid südamest means "I thank you from the heart." If you are saying "thank you" on behalf of a group of people, you can change Tänan "I thank" to Täname "we thank".
Aitüma: This is a less common way to say Aitäh. Another way to say it is Aituma. Keep in mind that it is an interjection, not a verb (although it ends in -ma and looks a bit like an infinitive form, it isn't, and it shouldn't be conjugated like a verb). Etymologically it comes from the phrase "aita Jumal" (God help), and the original phrase is better preserved in some of the southern dialect variants such as Aitjumma.
Ma olen sulle väga tänulik: This means "I am very thankful to you" or "I am grateful to you". Olen sulle südamest tänulik means "I am sincerely (from the heart) thankful to you." You can change either of these phrases to the formal form by replacing sulle with teile.
Suur tänu... eest: Use any of the words above to thank someone for something by using the word eest: Suur tänu abi eest, "thank you very much for the help," Tänan sinu kirja eest, "thank you for your letter," Aitäh hea nõu eest, "thanks for the good advice." You can also use et with with a verb or verb phrase: Olen sulle väga tänulik, et appi tulid, "I am very grateful that you came to help," or use the -mast form of the verb (ma-infinitive + st): Aitäh tulemast, "thanks for coming."
Ma ei leia sõnu, et sind küllalt tänada: This is a phrase you can use when you are extremely grateful for something. It means "I can't find the words to thank you enough." Mul pole sõnu, et sind tänada is "I don't have the words to thank you." In both of these phrases, sind can be replaced by teid when you need to use the formal form. Sa ei kujuta ette, kui tänulik ma sulle olen means "you can't imagine how grateful I am to you."
See on sinust väga kena: This means "it is very nice of you." The formal form is See on teist väga kena and you can also say See on sinust nii kena or See on teist nii kena ("it is so nice of you").
Pole tänu väärt: This is a very common way to respond to "thank you." It means "it's not worth thanks" (in other words, "don't mention it").
Palun: This is another response to "thank you," it is the same word that is used to say "please". You can also say Palun-palun.
Võtke heaks: This is another way to say "you're welcome" in response to "thank you." The informal form is Võta heaks.