Right now it takes 6 confirmations, which takes about an hour to verify a transaction. Will the rate at which transactions confirm become faster over time?
No. The average time per confirmation will always be about 10 minutes (but it's possible that the variance will be lower in the future). And the number of confirmations required to prevent double-spending by an agent with a given hashrate will also not change (6 is an arbitrary number reflecting a "low enough" probability for someone with a "high enough" hashrate).
What will change is services to bridge the gap. There will be split-key eWallets which cannot steal your money on one hand, and can have trusted green addresses on the other; and there will be insurance agents such as https://zipconf.com/ that absorb confirmation risk based on their model of the probabilities of double-spending.
Actually the amount of confirmations is seperate from verification. The more confirmations a transaction has the more certain a transaction is. The point where you say it is "verified" is simply the point where you trust it sufficiently.
When do you trust it? For a person this it's a feeling in your underbelly. How fishy is a certain deal? How trustworthy is the person transacting? How are his/her technical capabilities? How gullible is that person? For companies the equation is more exact, but can still be chosen. Can we take the loss if this was a trick? What is the chance that a transaction is fake after X confirmations and would it be more profitable to have someone cheat us every so often, if it makes our service faster? The perfect one is "Would an attack have been more expensive than the value of the transactions in the block in which they were created and those after?".
The notion of six is certain is a fallacy. However after the third confirmation this is a true 51% attack which is fairly hard to do. So six is abundantly secure.
http://bitcoinwatch.com/ now estimates the PetaFLOPS within the network's capabilities to be nearly 150. In January renting 240 Teraflops came at a price of 1280$ an hour. To get 75% of the current network's hashrate for half an hour one would have to spent 150000/240*1280*.5 = 400 000 USD's. So unless you're dealing with fairly huge numbers, 3 confirmations suffice. Edit: because the hash complexity doesn't grow every block it is 3 confirmations or 35 minutes, whichever takes longer. This should dampen the attacker's ability to burst a few blocks within a short period of time. The additional 5 to ensure sync. If you want that 400k limit you require an additional block to ensure the previous 3 were okay.
The gambling site satoshiDICE has an interesting take on this. They accept bets immediately, with zero confirmations, and pay out immediately if you win. This may seem risky, but there's a trick:
Part of your winnings are paid with the amount you sent as your bet. So if your bet was a double spend, and ends up not making it into the blockchain then neither do your winnings.
I thought that was quite a clever take on the whole question of how many confirmations is enough.
Oh, and to answer the question, each confirmation takes 10 minutes by design. If confirmations start coming faster than every 10 minutes then the network adjusts the difficulty of finding a block to slow down the confirmation rate. Similarly if confirmations are taking longer than 10 minutes then the network reduces the difficulty of finding blocks.
It may be in time that people start trusting transactions with 4 or 5 confirmations, or maybe they're get more cautious and start waiting for 10 confirmations. But that's nothing to do with the Bitcoin network - that's just people and their perception of risk vs. reward.
No, it probably won't. 6 confirmations, or 1 hour is considered a standard amount of time required for any Bitcoin-like crypto currency to ensure a transaction will not be reversed. The time to generate a Block will not change without changing the protocol (which is unlikely to happen), so one would still require an hour to generate those 6 confirmations.
However, even now you can see 0 confirmations transactions appearing in your client in seconds, and it is only up to you to decide when you consider that money secure. Recently 4 blocks were replaced by a miner with a big amount of hash power, meaning transactions in those abandoned blocks were vulnerable to double-spend attacks. However, such cases are rare.
Moreover, even right now some people are working hard on making the confirmation problems less bothersome to the users. Some want to establish some assurance from pools that would promise to include a transaction in their next block, other build bank-like systems that would process internal transactions without the need for blocks (currently you can do that between MtGox's accounts for example).
Generally, as a user one normally doesn't need faster transaction confirmations as double-spend attacks are not that frequent (not to mention detectable within a couple seconds in most cases) and one rarely needs to provide something to the receiver on the spot. It is mostly an issue for micropayment trasaction providers that need to clear an operation within seconds, not being able to wait even for 1 confirmation.