Some verbs are followed by -ing or to... with no difference in meaning:
begin, continue, intend, hate, like, love, prefer, start
She began to eat / eating her meal.
I love to eat / eating Chinese food.
I don´t like eating / to eat alone in restaurants.
Which dessert do you intend to order / ordering?
After the meal we continued chatting / to chat for a long time.
But when stop is followed by -ing or by to... there is a difference in meaning:
Please stop making that noise, it´s driving me mad! (= don´t continue)
Their mother told them to stop quarrelling.
We stopped to get some petrol and have some lunch. (= stop in order to)
I was half-way through my meal but I had to stop to answer the phone.
And when remember is followed by -ing or by to... there´s also a difference in meaning:
Did you remember to buy the lettuce for dinner? (= did not forget)
I don´t remember you asking me to buy a lettuce. (= I have a clear memory of it.)
You must remember to send Jill an invitation to the party.
I remember posting the letter yesterday evening after work.