What if I eat by mistake while fasting?

If you eat or drink by mistake because you forgot you were fasting, your fast is still valid. Stop as soon as you remember, spit out what's in your mouth, and carry on fasting. You don't need to make up the day, according to the majority of scholars.

What the Prophet ﷺ said

This is one of the kindest rules in fasting. The Prophet ﷺ said that whoever forgets while fasting and eats or drinks should complete the fast, because it was Allah who fed them and gave them drink (Sahih al-Bukhari 1933, Sahih Muslim 1155). In other words, it isn't counted against you at all.

This matches the dua Allah taught the believers at the end of Surah al-Baqarah:

رَبَّنَا لَا تُؤَاخِذْنَآ إِن نَّسِينَآ أَوْ أَخْطَأْنَا "Our Lord, do not impose blame upon us if we have forgotten or erred." Quran 2:286

What to do when you remember

Stop straight away. Spit out any food or drink still in your mouth.
Don't swallow it on purpose once you've remembered. That would break the fast.
Rinse your mouth if you like, and carry on with your day.
Don't feel guilty. Forgetting is human, especially in your first Ramadan when not eating is a new habit.

If it happens near the end of the day, it's still the same rule. See How do I prepare for my first Ramadan as a new Muslim? for more help getting used to fasting.

The Maliki view

The Maliki school says that in an obligatory fast like Ramadan, someone who eats by mistake should continue fasting but make up the day later as a precaution. The majority (Hanafi, Shafi'i and Hanbali) follow the plain wording of the hadith: no make-up needed. Either way, there is no sin and no extra penalty.

Mistake vs. wrong timing

Forgetting is different from getting the time wrong. If you kept eating after dawn because you thought it was still night, or broke your fast before sunset because you thought the sun had set, most scholars say you should make up that day, though some, like Ibn Taymiyyah, said no make-up is needed. Checking a reliable prayer timetable helps avoid this. See What time do I start and stop fasting in Ramadan?

Small things like your own saliva are never a problem. See Does swallowing my spit break my fast?

Should I remind someone I see eating?

Yes, gently. If you see a fasting friend eating by mistake, many scholars say it is good to remind them kindly, as helping each other in good. A quick "Hey, aren't you fasting today?" is enough.

Key points

Eating or drinking by mistake does not break your fast (Sahih al-Bukhari 1933).
Stop and spit out the food as soon as you remember.
The majority say no make-up is needed; the Maliki school says make it up as a precaution.
Eating because you misjudged the time is a different case.

Tip: In your first few days of Ramadan, put a small sticky note on the fridge or your water bottle saying "Fasting!" It's a simple trick that saves a lot of accidental sips.