What is an extremely basic asymmetric cipher that I can present at the pub?
If you want to present public key cryptography to your parents or friends, then I suggest you follow some guidelines. First, don't talk about specific functions, nobody cares about SHAxxx, keep your talk conceptual. The problem solved by public key cryptography is allowing two parties who never met before to securely exchange information in a public channel. Do not confuse this with other applications that are achieved by public key encryption, such as digital signatures or authentication.

An illuminating simple illustration of this that I have seen is the following. Suppose that two parties have a bucket and a number of colors, and they want to agree on a specific secret compound. They are allowed to send each other a bucket with a certain compound, but anything sent is exposed to an evil adversary. The underline assumption is that mixing colors is easy, but decomposing a mixture to its ingredients is hard. This assumption (which is easily connected to the formal side of things) allows you to achieve their goal. Pick a public color $P$, and allow both parties to pick a secret color, let us denote them by $S_1,S_2$. The first party then sends a bucket of a mixture of $P,S_1$. Similarly, the second party sends a bucket containing a mixture of $P,S_2$. Finally, each side adds his private color to the mixture he received, and now both parties have a mixture of $P,S_1,S_2$, which (by our assumption) remains secret from any eavesdroppers. A formal version of this is the Diffie-Hellman key exchange protocol, which relies on the hardness of discrete log.

Now, my understanding of the bitcoin protocol is limited, but when talking to your parents about it, I see no reason to go into cryptography. Most of the popular explanations I have seen unnecessarily delve into implementation details, and start talking about finding a preimage of a hash such that the result would have $x$ number of zeros, loosing sight of the actual problem. As before, I suggest to keep the talk conceptual. What problem is the bitcoin trying to solve, and why isn't this trivial? To my understanding, the bitcoin solves the problem of maintaining a distributed ledger, specifically avoiding the double spending problem. The motivation is to avoid having a central authority, and this can be achieved by allowing each participant to approve a transaction (instead of giving this authority to the bank). To avoid adversarial participants approving their own invalid transactions, the concept of proof of work is introduced. POW uses the hash as a black box, so you can avoid mentioning specific candidates.