An analogy that I find helpful for newer people is to think of the:
- Symmetric algorithm as a physical key and an associated "symmetric type of" safe
- Asymmetric algorithm as both
- an open padlock (your public key) that also has an associated "asymmetric lockbox"
- the physical key that unlocks the padlock (associated private key)
- The Password Based Key Derivation Function (e.g. PBKDF2) as being a "combination safe" dial mechanism (that uses the symmetric safe "underneath" it)
You first generate your asymmetric key pair. The public key can be given to anyone. It is like an open padlock. If someone writes a message to you and puts it in an "asymmetric lockbox," they can lock the lockbox with your public key by "closing the padlock," and then even they can't open it again without the private key. However, the message itself isn't actually what is protected with this padlocked lockbox, for although it is an awesome tool and effectively (currently) impossible to break into when used properly, it really isn't that strong or sophisticated in numerous other ways (it is a bit recalcitrant of a padlock that is slow to force closed, and the lockbox is known to be drillable by powerful quantum computers, and you can't nicely fit much into it). More on this later.
Depending on the scenario, mostly whether the key pair is long term or ephemeral, you may want to locally encrypt your private key symmetrically. This is typically done with a symmetric algorithm keyed with a PBKDF (I use PBKDF generically; most don't seem to unfortunately. I don't know what else to call the generic construct). You provide a password to the PBKDF, and it outputs a key. You can think of it as being like a combination safe (when it is coupled with a symmetric algorithm that actually encrypts things with it; the PBKDF is just the dial mechanism; it uses the symmetric safe for security of what is stored). The U.S. courts see it as being analogous to a combination dial because what is needed to unlock it is stored in your mind instead of in a drawer like a regular key is (such regular keys can also be used: they would be called keyfiles in this context, and have less legal protection in the USA because of not being stored in one's mind, but I digress). Now you only have your private key in plaintext when you need to use it; upon needing it, you type in your password (dial in the combination, by analogy).
So, when people want to send you a message, they actually put it in a very strong key based symmetric safe that is well oiled, quick to open and shut, resists "quantum drills," and can hold a lot in it. But how can they now ship this securely to you? If they locked the key in the safe, you couldn't use it to open the safe. If they shipped it along outside of it, anyone who intercepted the shipment could open it and read the message to you. This is where the open padlock comes into play. They put the key (called a "session key" and generated randomly per message) into an asymmetric "lockbox" and lock it with your padlock (after some effort to get the thing to shut -- it isn't the most friendly locking mechanism, nor the strongest, but eventually it can be forced shut).
Note that the padlock is a magical replicating one, and that you effectively have unlimited of them in the open state.
Now they can send you both the lockbox and the safe.
When you get the lockbox and the safe delivered to you, first you use your combination dial (type your password in) to unlock your local safe (decrypt your locally symmetrically encrypted private asymmetric key) to be able to take your private asymmetric key out of it. Second, you use this private asymmetric key to open the closed padlock on the lockbox, which allows you to take out the session key. You then use the session key to open the safe and get your message out of it.
That neglects authentication and other steps, and it is clearly an analogy; however, I believe it is a highly useful analogy for newer people. The use of the open padlock is obvious: how else can you have the ability to lock stuff for someone else to open? If you use a regular key lock, you both need the key, but if you don't both have it then how can you securely send it to the person who needs it? Certainly not outside of protection, for if you were to do so (in our simplified analogy not concerned with polymorphism and the like, anyway) you may as well have saved on shipping costs by not sending the safe or lockbox in the first place.
The key locked safes just have nicer mechanisms in and of themselves, and are nicer safes, so they are used when they can be, but you simply need the open padlock style mechanism so that you can send the key to the key safe, in the smaller and less secure lockbox. Both are sent in the same "shipment," with your message itself kept in the superior key locked safe because there is no need for anything but the session key to be protected with the comparatively not so good (but crucial) padlock mechanism and lockbox.