The other answers contribute pieces to this puzzle, but there is a bit of detail which is omitted which I think might clarify things a bit more.
As mentioned, for a given security there is this thing call the "order book", which consists of orders to buy and sell the security for specific prices. A such, the order book consists solely of limit orders, with the caveat that the order book is not limited to normal buyers and sellers (traders), but will also contain orders created by market makers. (I would expect this to be almost entirely algorithmic these days). So looking at the bid and ask size at a given moment in time is, practically speaking, useless for predicting the short time price movement of the stock, because individual traders can withdraw or adjust their limit orders at any time, and so can market makers, much faster. High frequency traders would also fall into this category, as a limit order could be placed that only exists on the books for a fraction of a second before it is withdrawn.
Except for a few specific cases, all the action actually takes place due to market orders. The few specific times where this is not true is at times like market open, where there could be bids at or above some asks, and asks at or below bid some bids. These get crossed (matched up) to produce the market price at the open, at which time the highest bid is some non-zero amount below the lowest ask and bid prices decrease from there, while the lowest ask starts at some non-zero amount above the highest bid and asks increase in price from there.
Another specific time where a limit order would cause an immediate trade in the market is if, at the time when the limit order is placed, there is a corresponding order on the books that it can match again. For instance, if I place a limit order to buy at 100 and the best ask price is only 99, I would pay the ask price of 99. When I say "at the time" I mean it somewhat loosely, as there is a notion of ordering, so if I placed the buy order a few milliseconds after another trader placed an identical order, their order would be ahead of mine and execute first, at which point the ask price might have gone up above my limit in which case my order would go into the book. For the point of looking at bid and ask size, we can consider such a limit order the same as a market order - it's not going to show up on the order book for any appreciable amount of time, it's just going to execute as if it was a market order.
When a market order is placed, if it is a buy order, then it's going to be matched against the current best ask until the order exhausts the ask size. (I think it's possible that high frequency trading or other shenanigans like "price improvement" might in some cases actually distort this somewhat, and also different trades can happen on different venues so order routing can affect price (the complete "order book" is actually an aggregation), but in principle this is what happens.) If the order size is less than the ask size, the ask size will decrease by the order size. If it's greater, that line of the order book will be removed and the next higher ask will become the "market price" and the order will start getting matched against that until it is exhausted, etc. When a market sell order is placed, the same thing happens except in reverse, with the bid price. Of course, immediately after the order is executed other participants might make changes to the book again, for instance, a market maker could have an algorithm that in a certain circumstance tries to always keep a certain amount of stock at a certain bid, so you well could sell 100 shares into a bid depth of 1000 and see no change in bid depth, because the depth was "replenished" so to speak.
Note that this means that a very large buy or sell order could "blow through" the existing supply or demand of the stock and cause a large swing in the price. To avoid this, various algorithms may be employed such as parcelling up a large order into smaller pieces. So you could see the same behavior of bid size always seeming to stick around 1000 shares because somebody is trying to sell a large block at that price but is only offering 1000 shares at a time. What size the algorithm picks, as well as any other participants at the price level, can affect what the actual size is.
Bottom line is that the existence of algorithms both for buying and selling stocks in a less disruptive (and hence more profitable) manner and by market makers is often going to negative any useful information that bid and ask size could provide.