What happens if you lose the private key of a Bitcoin address which holds some Bitcoins? Can the Bitcoins ever be spent if you don't have the secret key? The answer in both cases is, you guessed it, that those BTC are lost forever. Losing the private key puts you in the same situation regarding those coins as anyone else on the Internet: it's impossible, for all practical purposes, to guess the information needed in order to sign transactions originating from that address. If that were possible, then anyone could guess other addresses' keys and spend those coins as well.
But, if it's any consolation, if you've lost the secret key of a BTC address you are definitely not alone. In fact, it is estimated that over 23% of all Bitcoins ever created have already been lost!! How is this possible? Why didn't all these folks take better care of their secret keys?
In some cases they aren't considered lost, but just haven't been spent yet. Such is the case with the legendary Satoshi Nakamoto's coins. It is estimated that, during the early days of Bitcoin, Satoshi mined approximately 1 million coins which remain untouched. (Of course, if any of his suspected addresses were ever used, that'd surely cause a stir in the Bitcoin community.)
Rumours have it that Hal Finney and Satoshi (same person?) designed Bitcoin to have 21 million coins so that the public could have access to 20 million and they could test the system using the first 1 million coins. This wasn't a naive decision, as Finney knew the value that each BTC could reach in the future were it to be come successful. The first 1 million BTC by themselves represent approximately 5% of the maximum available supply. To mine the first million coins, Satoshi would've spent 139 days mining all by himself, considering a rate of 50 BTC every 10 minutes.
In the early days of Bitcoin people really didn't take it very seriously. Some folks infamously bought pizzas using large sums of Bitcoin, others threw away their hard drives containing coins and some people simply installed Bitcoin, left the node mining on its own and later forgot about it. Remember, back in the early days, simply leaving the Bitcoin Core software running would generate BTC (before professional mining became a thing).
All the Bitcoins for which the secret keys were lost will likely never be recovered. Or maybe they will, in 20 years or so when quantum computers are able to break the elliptic curve cryptography used in present day BTC. But we suspect that Bitcoin will have adapted to protect itself to quantum computing algorithms by then.