One cup/glass/bowl/spoonful — a single serving or measure of a liquid or similar item.
コーヒーを一杯ください。
Please give me one cup of coffee.
味噌汁をお椀に一杯よそった。
I ladled one bowlful of miso soup.
レシピには砂糖を大さじ一杯入れると書いてある。
The recipe says to add one tablespoon of sugar.
A single drink of alcohol (one drink/one glass).
乾杯のあと、もう一杯いかがですか？
After the toast, would you like another drink?
仕事帰りに同僚と一杯やった。
I had a drink with a colleague after work.
Full or brimming (with); crowded or packed — used to say something is filled to capacity.
箱は古い本でいっぱいだった。
The box was full of old books.
駅のホームは通勤客でいっぱいになった。
The station platform became packed with commuters.
彼女の顔は笑顔でいっぱいだった。
Her face was full of smiles.
Completely; to capacity; as much as possible — adverbial use (いっぱいに).
瓶に水をいっぱいに注いだ。
I filled the bottle to the brim with water.
スケジュールは予定でいっぱいだ。
My schedule is full of appointments.
A lot; many; much — casual adverb indicating a large quantity or degree.
写真がいっぱいあるから選ぶのに時間がかかる。
There are so many photos that it takes time to choose.
彼は経験がいっぱいある。
He has a lot of experience.
All of ...; the entire ... — used to indicate the whole/entirety of something.
町中一杯に桜の花が咲いている。
Cherry blossoms are blooming all over the town.
部屋一杯に香水の香りが広がった。
The scent of perfume spread throughout the room.
Counter: one (for squid, octopus, crab, etc.) — used as a counting unit for certain seafood.
市場でイカを一杯買った。
I bought one squid at the market.
たこを一杯調理するのに時間がかかる。
It takes time to prepare one octopus.
Counter: one (boat) — an older/counting usage indicating one boat.
港に小舟が一杯停まっている。
One small boat is moored in the harbor.