As a non-English-native person, I somehow find it difficult to understand the following: When an account is said to be debited, what does this mean? How does this differ from an account that is credited? Both seem to be checking(withdrawal-possible) account, and I am having difficulty distinguishing these two. How do they differ in some math-quantity understanding in economics?
Strictly speaking the terms arise from double entry book keeping terminology, and don't exactly relate to their common English usage, which is part of the confusion.
All double entry book keeping operations consist of a (debit, credit) tuple performed on two different books (ledgers).
The actual arithmetic operation performed by a debit or a credit depends on the book keeping classification of the ledger it is performed on. Liability accounts behave the way you would expect - a debit is subtraction, and a credit is addition. Asset accounts are the other way around, a debit is an addition, and a credit is a subtraction.
The confusion when dealing with banks, partly comes from this classification, since while your deposit account is your asset, it is the bank's liability. So when you deposit 100 cash at the bank, it will perform the operation (debit cash account (an asset), credit deposit account). Each ledger account will have 100 added to it. Similarly when you withdraw cash, the operation is (credit cash, debit deposit). However the operation that your accountant will perform on your own books, is the opposite, since the cash was your asset, and now the deposit account is.
For those studying math, it may also help to know that double entry book keeping is one of the earliest known examples of a single error detection/correction algorithm.
I'm not sure if this is your point of confusion, but when an account is said to be debited (or credited), the words "debited" or "credited" are not referring to a type of account (such as "checking"). They are referring to an operation that is performed on an account. The same account can be credited at one time and debited at another time.
The credit and debit terms here is, talking from bank's point of view (shouldn't be a surprise, banks are never known to look at things from the customers' POV ;)).
In accounting, a liability (loans, owners capital etc) is a credit balance and asset (cash, buildings and such) is a debit balance. Your account is a liability to the bank (in accounting parlance that is because they owe you every single penny that is there in your account, btw, in literal parlance too if you really make their life harder ;))
So when the bank accepts money from you, they need to increase their asset (cash) which they will debit (higher debit balance for asset means more assets), and at the same time they also have to account for the added liability by "crediting" the deposited money into your account.
So when bank says they have credited your account, it means you have more money in your account.
Now, if you transfer money from your account to another, or make a payment through your account, your account will be debited and the beneficiary account will be credited(bank's liability towards you reduces)
More or less what everyone else said here... but hey, I could also take a swipe at banks ;))
The terms debit and credit come from double-entry book-keeping. In this system, every transaction is applied against two accounts: it debits one and credits the other by equal amounts. (Or more technically, it affects two or more accounts, and the total of the credits equals the total of the debits.)
Whether a debit or a credit adds or subtracts from the balance depends on the type of account. The types of accounts were defined so that it is always possible to have these matching debits and credits. Assets, like cash or property that you own, are "debit accounts", that is, a debit is an increase in the balance of the account. Liabilities, like money you owe, are "credit accounts", that is, a credit is an increase. To get into all the details would require giving a tutorial on double-entry book-keeping, which I think is beyond the scope of a forum post. By a quick Bing search I find this one: http://simplestudies.com/double-entry-accounting-system.html. I haven't gone through it so I can't say if it's a particularly good tutorial. There are plenty of others on the Web and in bookstores.
Note that the terminology can be backwards when someone you're doing business with is describing the account, because their viewpoint may be the opposite of yours. For example, to me, my credit card is a liability: I owe the bank money. So when I post a charge, that's a credit, and when I pay it off, that's a debit. But to the bank, my account is an asset: the customer (me) owes them money. So to the bank, a charge is a debit and a payment is a credit.