The bitcoin mempool is where unconfirmed transactions are stored.
It's a set of all the transactions that have been broadcast but have not yet been added to a block by a miner.
Every node has its own transaction relay policy so there is no definite mempool, but different mempools for different nodes.

Legacy means that you will be using addresses of the form 1... 
Segwit means that you will be using addresses of the form 3... or bc1... (one is for P2SH nested segwit and the other is for native segwit). 
Note that 3... addresses are for P2SH addresses in general and are not just for segwit.
bc1... addresses are for segwit specifically but not all wallets support it yet.*

It checks the signatures of the transaction against previous blocks.

Yes, Bitcoin does have a solution and it is called the proof-of-work algorithm.
So to alter "the message" (a block of transactions in our case) as in the classic Byzantine Generals problem and to make it look authentic by guessing a new nonce, one would need to use more Hash Power than the entire network.
Since the current Hash Rate is around 23 million TH/s, to achieve that they would need to buy 1.5 million of Antminer S9.

I actually did this kind of backwards. I had to 
write all the code before I could convince myself that I could solve every 
problem, then I wrote the paper. I think I will be able to release the code 
sooner than I could write a detailed spec. [i]-Satoshi[/i]

Satoshi implies that handling the amount of transactions that Visa processes should not be a problem in time because it was expected that computing power would keep pace with network and transaction growth.
There were concerns with blocksize limits at the time because the bigger they got the more power that would be required with both upload steams (to broadcast to the network) and computing power (to process the transactions). 
The blocklimit was set so that mining operations could not out-power single person operations.

Satoshi's emails re transaction processing imply that higher block limits were anticipated and were an expected development of bitcoin.
This is controversial as a number of people see off chain scaling as a better method instead of increasing block size. 
At 150,000,000 transactions a day that makes 1,041,667 per block (given it can be mined in 10m), and with a transcation size of around 1kb it would need to be just over 1gb.
My understanding is that upload/download speeds should be able to keep pace with this size.

A database is online, but it is kept there by a server computer, so it is centralized by definition.
If I have a database, I control it and I can manipulate it as I see fit. It can be hacked or changed by others.
A blockchain is decentralized, and it is on hundreds or thousands of computers, based on the popularity of the blockchain.
Some, such as bitcoin blockchain is very popular due to the high price, and some others are not so. 
Blockchain is an immutable public ledger and it cannot be hacked, modified, stolen or manipulated. 
If you own 51 percent of the blockchain though, then it could be a problem as they can all pool and do something nasty with that.

If I had 51%, I could mine a chain of blocks in which I transfer all my coins to my personal wallet. 
I'd mine this chain about 10 long, but not tell the rest of the network. 
At the same time, I convert all my coins to dollars on the exchange and withdraw them. This happens on the normal blockchain.

After my withdrawal has gone through. the normal blockchain is about 9 long, while my blockchain is 10 long. 
I announce all my blocks to the network, and lo and behold, the network confirms I am right.

But dollars can't be reverted! So the exchange takes a loss.

Instead of the exchange, I could do this with buying anything for bitcoins. 
If this happens a few times, it will probably kill bitcoin, or at least hurt the trust in the system severely.*

With one confirmation you are vulnerable to the 51% attack. There could also be a miner with a lot of hashing power who could get a couple blocks in a row, so three confirmations removes most of them.

With six confirmations it is essentially mathematically impossible for an attacker with less than 51% of all mining capacity to get six blocks in a row. and still surpass the longest block chain. With 51% or a lot more than 51% the attacker can get six confirmations by creating a parallel blockchain in which only transactions approved by the attacker get included in blocks.

Therefore 6 confirmations are needed to prevent the attack from attacker.

Bitcoin days destroyed is another way to measure bitcoin trading volume, in a more significant way.

This idea was created because 1 bitcoin could be sent thousand times a day, to create a false high volume.

The idea behind Bitcoin days destroyed is to consider in the trading volume equation the number of days each bitcoin has not moved from that address.

So if you have 1 bitcoin that has not moved in the last 100 days, when it move to another address it would have the same weight in the volume as 100 bitcoin that were transferred yesterday.

This equation would give a more precise value for bitcoin volume and economic activity as it cannot be manipulated by whales.

Double spend is to spend the same coins in 2 different transactions. It's mostly attempted when the 1st transaction doesn't have any confirmations yet, because it becomes much harder to successfully do a double spend after each confirmation.
Double spend is usually attempted when someone makes a transaction by mistake, when a transaction gets stuck because of a low fee, or of course for fraud.
There are several ways to attempt a double spend, what I know of them:
Including a higher fee in the 2nd transaction (the one you want to get it confirmed faster), asking a miner to include your transaction in the next block, these two methods work for unconfirmed transactions.
Double spend can also happen in 51% attacks or similar attacks where the attacker has high hash power. In this case the 1st transaction could have a couple of confirmations and still get reversed. Though this kind of double spend is not worth it unless the amount of transaction is so high.