Now here’s a topic that’s so basic I can’t believe I haven’t done it yet. Sheesh.
The Arabic verb “to thank” is شَكَرَ (shakara), which also means “to be thankful.” After last time you should be familiar with the idea that you can take this root and derive other words from it, yes? Well, شكر has a pretty limited set of applications but the vocabulary is useful nonetheless.
“Thank you” is شُكراً (shukran). “Thank you very much” or “thanks a lot” is usually شُكراً جَزيلاً (shukran jazīlan), though colloquial variations do exist. Technically the full phrase is شُكراً لكم (shukran la-kum, “thanks to you”) but the specification “to you” is almost always omitted except in very formal talk.
That structure at the end, the اً (-an) is something we should talk about. It’s called a تَنوين (tanwīn), and it is the indefinite marker (i.e., it means you’re talking about “a” thing and not “the” thing). It’s written as a doubling of the final short vowel (which determines case: ُ marks the nominative, َ marks the accusative, and ِ marks the genitive–we can maybe cover this at some point in more detail). When a noun in whichever case is indefinite, it ends with a تنوين in the appropriate case, and the effect is to add an “n” sound after the vowel. For the فَتحة (fatḥah), however, you can only add the تنوين directly to the feminine marker (ة); for all other letters you have to also add an alif (ا) and then tack the تنوين on to that. The reason I haven’t really mentioned this before is that, in everyday speaking, you almost never articulate the case endings to anything (nor would you usually encounter them in writing since short vowels are usually omitted), but in the case of شُكراً it is always spoken.
Other useful words that derive from شكر include the passive participle مَشكور (mashkūr), “worthy of thanks” or “praiseworthy,” the active participle شاكِر (shākir), “thankful” or “grateful,” and تَشَكُّر (tashakkur), which also means “gratitude” or “thanks” but is more emphatic and is really important when we come to Turkish (and Persian, though it’s less important there). There’s also مُتَشَكِّر (mutashakkir), which also means “grateful” or “thankful” but doesn’t come into play much in Arabic (it’s more important in Persian).
The response to شُكراً, usually, is عَفواً (ʿafwan), and note that this also vocalizes that indefinite ending. The verb عَفا (ʿafā) means “to excuse” or “to pardon,” and the expression عَفواً can be used when you want to say “excuse me” or “pardon me,” but it also means “you’re welcome” in a “don’t mention it” sense.