Limit orders placed above/below best bid/best offer

First you need to understand that a limit order is simply an order which has a limit on the price i.e. Maximum price for bids and minimum for ask. The trade won’t take place above or below these limits for buying and selling respectively.
A marketable limit order is when you set the limit above the prevailing bid prices (when buying). Basically means you’re willing to pay the most at an auction so you have the best chance of getting it. Marketable order is also applicable when you set the limit below the ask prices (when selling). Hence marketable orders are most likely to be executed.
Behind the market limit order is when you set the limit price below prevailing bid prices (when buying). Basically means that you’re willing to pay less than the market price so no one’s going to give a damn about your lousy bid, hence the only chance you have of getting the security is if the price falls and suddenly your bid is the best. Use the same logic with ask prices (when selling).
I’m not sure what at the market is but my guess is it’s when your bid exactly matches the bid price in the market and ask price matches the ask price in the market. Again this may not get executed because the seller/buyer may choose someone who has bid/asked for a greater volume.

Hope you understood. Others please correct me if I’m wrong since I have no real world experience and this is from what I know

@RoyD‌ you are right, even about “at the market”, however at the market is almost always executed since it is executed at market price (as long as there is a counter-trade also happening).
@simply_complex2
“For example, my book says a limit buy order placed at best bid is at the market, but wouldn’t you want to compare the limit buy order to the best ask since that’s the lowest price a broker would sell you the security?”
X placed an order to sell stock ABC for $100 or below (best ask)
Y placed an order to sell stock ABC for $101
Assume market price of ABC at the moment is $101.

Now, you place an order to buy stock ABC. If you placed your order at best bid, the value would have to be $101 and your order would get executed at market price (hence at the market) immediately. Since the market price is $101, if you place an order at best ask ($100) your order would be placed in the queue pending execution.
Remember the definitions of best ask and best bid:
best ask: lowest selling price
best bid: highest buying price

I hope you understand the above. If not, consider this extreme case: assume that someone places a sale order for ABC at $10 (market price is $100). You, just for kicks, place a buy order at $10. If such orders start getting executed, there won’t be any meaning to trading stocks at all – it will cause pandemonium in the markets. Hence, the above statement is valid (and is in fact how trading orders are satisfied).

Hope that clears your doubt.