- What is the main difference between savings and checking account?
A checking account almost never earns any significant interest. A checking accounts often does not have any limits in terms of how many times you can draw funds from it. A checking account often comes with a debit card allowing you to pay online, draw cash from an ATM machine etc.
A savings accounts has much higher yields, so you should be getting a decent interest.
Unfortunately in the current climate this is not always true (especially not with the big banks) so you may want to look into a high-yield savings account. A savings account is often limited in terms of number of transactions, meaning you can't constantly draw funds from it, it must be stable. A savings account often does not come with a debit card.
- Will I be able to use my saving accounts to perform transactions online? (Pay or receive payments online?)
No, a savings account should not be used for regular transactions. It's an account to park your money for a medium/long time. Understand that banks loan out the money in your savings account to third-parties, so if it would constantly fluctuate, they can't have this money available to others. In return of you parking your money with your bank, you should get a nice return (interest).
- Can a third party deposit to my account? (Say I'm selling something and I ask him/her to just deposit the payment to my account?
Yes, but it's not common. Assuming you are from USA, passing banking data (account number and routing number) to third-parties is not safe. In Europe it's totally safe to share your account number to accept money.
- Do you have to pay monthly fees for having a bank account?
Depends, some banks do charge fees, some don't. Often there are fees when you're not using the account (no transactions), or when you don't have a certain minimum in an account.
- Which kind of bank account will be more ideal for a student with no job?
Assuming you are American (please specify this information clearly in future) I would look into an "internet bank", like Ally. They don't have many fees and they have an excellent high-yield savings account. They also give you a debit card. Disadvantage is that they don't have physical branches.