Amiability or friendliness — a pleasant, sociable attitude (used in phrases like 'friendly' or 'not friendly').
あの店員さんはいつも愛想がいいので気持ちがいい。
That clerk is always friendly, so it's pleasant.
初対面の人には愛想良く挨拶した方がいい。
It's better to greet a person you're meeting for the first time politely and amiably.
彼女は愛想が悪いと言われることがあるが、内心は優しい。
People sometimes say she's unfriendly, but she is kind inside.
Fondness or affection for someone — liking or affection directed toward a person (as in 'to lose affection').
彼女の無責任な行動に彼はついに愛想を尽かした。
He finally lost affection for her because of her irresponsible behavior.
長年の信頼関係が壊れ、愛想が薄れていった。
Years of trust were broken, and the affection gradually faded.
Polite remarks, civilities, or flattery — social niceties and small talk used to be pleasant to others (sometimes as 'o-aisou').
初対面なので軽く愛想を言って場を和ませた。
Since it was our first meeting, I made some small talk to lighten the mood.
営業マンは愛想の良さで商談を進めた。
The salesperson advanced the deal with his pleasant manners.
Hospitality or special treatment — entertaining or giving preferential treatment to guests (often expressed as 'o-aisou').
常連だから店主がお愛想してサービスしてくれた。
Because I'm a regular, the owner gave me some special service as a favor.
観光客にはお愛想で無料の試食を出していた。
They offered tourists free samples as a courtesy.
The bill or check — a colloquial term (often 'o-aisou') used to ask for the check in restaurants or bars.
そろそろお愛想をお願いします。
Could we get the check, please?
飲み会で『お愛想』と言って帰った。
He said 'o-aisou' at the party and left (meaning he paid and left).