Good questions. Inline:

So the value of a bitcoin is the agreed value of a distributed ledger?

The value of a bitcoin is whatever it can be exchanged for. As the network becomes more utile for all participants, that value goes up. As it becomes harder to use, or less broadly trafficked, the lower it goes. Confederate currency was worth something when the Confederacy was new. Confederate currency after the war? Worthless.

Is ether just another currency like bitcoin?

No. As above, Ether is the value of processing cycles on Ethereum, which is a Turing-complete network capable of executing smart contracts (effectively software). The more things Ethereum can be used for, the more in-demand Ether becomes. Ether (the marker, representing work time on the system) is cryptocurrency but it's cryptocurrency intrinsically tied to an actual thing (work time on the system, which is Ethereum).

Can ethereum work without bitcoin?

Ethereum and Bitcoin are wholly independent of each other.

Looks like ether skyrocketed over the last couple of months, what happened?

Public awareness, for one. The value of any currency is tied to the number of people using it - the dollar is strong, for example, because it's effectively the reserve currency of world trade. That awareness has increased for a few reasons:

(The SEC later agreed to revisit, which boosted it right back up)

Additionally, Ethereum looks like they're going to start switching over from proof-of-work to proof-of-stake. What you need to know at a basic level is that proof-of-stake effectively means an end to mining. The amount of Ether will grow much slower. The nodes supporting the network will be working much less. Proof-of-stake effectively means "okay, everybody, we're done with the investment stage."

How do these cryptocurrencies jump from their current obscurity to more mainstream adoption?

Doesn't really matter. these organizations are looking at implementing the Ethereum blockchain to interact internally and with other Alliance members. You don't ever need to use Ethereum at all and you'll still be using Ethereum. It's like the United States: we don't use the Metric System, but since inches and pounds and gallons are all defined in terms of Metric units, the US is on the Metric System. You can continue to use dollars for the rest of your life but if you have a transaction between your UBS account and your buddy's State Street account (for example), you're using Ethereum. And, as per the article, if Russia decides to go Ethereum, it will certainly signal other participants.

Is there a specific niche they were made to fill alongside existing currencies?

Bitcoin is intended to facilitate the exchange of value between individual participants in a way unmoderated by any central bank. That's the core concern: your money is worth whatever the market says your money is worth, not what the Federal Reserve says it's worth. It does this by using the distributed ledger to provide a consensus of exchange.

Ethereum is intended to facilitate the exchange of value between any participants in a way in a way unmoderated by currency. That's the core concern: your labor/goods/services/other can be exchanged for any other labor/goods/services/other without passing through a transaction layer. The implications of this are still pretty fuzzy, but could be revolutionary.