With public-key crytography, is private-key crytography still useful? It seems that, with public-key cryptography, there is no need to use private-key cryptography. But obviously, both public-key cryptography and private-key are taught in cryptography classes. Can anybody let me know when private-key crytography is useful in practice? If so, can it be replaced with public-key cryptography?
Can anybody let me know when private-key crytography is useful in practice?


Symmetric techniques tend to be significantly faster than asymmetric techniques. Encrypting data with AES and an appropriate mode of operation can yield throughput that is magnitudes of order greater than what the fastest implementations of RSA could possibly deliver. This difference is so great that encrypting any sizable amount of information with only RSA would take a prohibitive amount of time and be unusable in practice.

So symmetric cryptography techniques are useful pretty much everywhere in practice, from encryption of bulk data at rest (e.g. full-disk encryption) to encryption of data in transit (e.g. TLS)


  If so, can it be replaced with public-key cryptography?


You could simply use asymmetric cryptography for everything. But there are much more intelligent ways to do things. 

You can, for example, generate a regular symmetric encryption key and use that to encrypt your data quickly, and then use public-key encryption to secure the key. This nets the following advantages:


You can use a new secret key per encrypted message
But you only have to manage long-term control of a single private key, instead of $N$ secret keys for $N$ messages.


This highlights what Public-key cryptography is really good at: Key management. 

Using public-key encryption to directly encrypt bulk data won't work out well in practice, because that's arguably not what the tool is supposed to be used for.