The Wikipedia article on ACPI contains the details about the different power savings modes which are present in modern PCs.
Here's the basic idea, from how I understand things:
The basic idea is to keep the current state of the system persisted, so when the machine is brought back into operation, it can resume at the state it was before the machine was put into sleep/standby/hibernation, etc. Think of it as serialization for your PC.
In standby, the computer will keep feeding power to the RAM, as the main memory is volatile memory that needs constant refreshing to hold on to its state. This means that the hard drives, CPU, and other components can be turned off, as long as there is enough power to keep the DRAM refreshed to keep its contents from disappearing.
In hibernation, the main memory will also be turned off, so the contents must be copied to permanent storage, such as a hard drive, before the system power is turned off. Other than that, the basic premise of hiberation is no different from standby -- to store the current state of the machine to restore at a later time.
With that in mind, it's probably not too likely that going into standby or hibernate will cause problems with tasks that are executing at the moment. However, it may not be a good idea to allow network activity to stop in the middle of execution, as depending on the protocol, your network connection could timeout and be unable to resume upon returning the system to its running state.
Also, there may be some machines that just have flaky power-savings drivers which may cause it to go to standby and never come back, but that's completely a different issue.