The advice in "debit advice" and "credit advice" is not advice in the sense of telling you what you should do. Instead, it is information advising you that a transaction has been made in your bank account. The "debit" and "credit" references are the way round that you might expect with a bank account, unlike in accounting where they can have the opposite meaning. A "debit under advice" is something different and refers to a bank penalty charge.
The term "advice" can be particularly confusing when used by your bank. Banks can sometimes give you advice in its more common meaning, meaning they give you suggestions of what financial decisions you should make, although regulations heavily restrict this advice. More commonly advice from your bank is simply confirmation of what has happened. The most common example is when you use a cashpoint and are offered an "advice slip," which is really just a receipt for your withdrawal.
Debit and credit
Debit and credit have their "normal" meaning when used in relation to your bank account: that is, a debit reduces your bank balance and a credit increases it. Of course, if you are overdrawn, a debit will reduce your balance but will increase the (overdrawn) amount, with the opposite holding true for a credit. A further source of potential confusion is that if you have a debit transaction while overdrawn (or that takes you overdrawn) you are using a credit facility. Even in this situation, debit and credit advice still relates to reductions or increases in your balance respectively.
As well as advice slips from cashpoints, you may get a debit or credit advice in a range of ways. A common example is the monthly printed statement that lists all the transactions during the month, though online accounts often let you view statements for custom periods. Some banks may issues debit or credit advice for individual transactions. For example, if you make a payment through a telephone banking system, you might get a text message confirming the payment has gone through; this would be a debit advice. Strictly speaking a message from a third party confirming it has paid you or taken your payment isn't a credit or debit advice as it doesn't come directly from a bank
Debit under advice
Occasionally your bank account may list an item as "debit under advice." This term appears not to be used consistently among banks so you will need to check with your bank to see what it means. In some cases it can refer to a charge imposed by your bank after it refused to make a direct debit or standing order payment because you did not have enough money in your account. In other cases it can refer to a charge (also called a debit card abuse fee) imposed by your bank when you didn't have enough money in your bank for a payment, but the bank was forced to make the payment anyway. This could happen if you wrote a cheque with a guarantee card.