What is a soft-money research position?
It's actually a little bit more complicated than "soft money" vs. "hard money," but the basic idea is that any scientific position can be categorized on the basis of where the money to support it comes from.

In general, "soft money" refers to positions where the funding source is based on external grants and contracts.  As long as there is a sufficient supply of such money that a person can obtain or that somebody else is willing to devote to their support, they are employed.  If that money dries up, however, the position dries up with it, becoming part time or disappearing entirely.

In contrast, "hard money" positions are not directly project-dependent, but instead are created for a purpose by an organization.  Termination from a hard money position only comes either for cause or due to the organization deciding to remove the position and not shift the person to another position.  They are often more secure (thus the notion of "hard"), but not necessarily.  Tenure-track faculty posts are typically seen as the ultimate in "hard" money: they are ultimately supported by tuition, and even if the students stop signing up for a department, it is typically slow enough that universities tend to prefer to allow departments to age to extinction rather than fire faculty.  A corporate research post is also often "hard money" because it is created by fiat in a company's strategic plan, rather than being dependent on external funds.  It is much less secure, however, because it can be removed by fiat as well.

Where it gets complicated is that various different sorts of "hard" and "soft" positions have different levels of vulnerability.  For example, some "soft" positions are effectively amortized over a large group or include bridge funding, such that they may be nearly as secure as a tenure-track position.

It's a good first-approximation, however, in talking about the challenges of academic fundraising and career management.  The basic rule of thumb is: a PI in a "soft" position needs to raise money to pay their own salary, while a PI in a "hard" position does not.  Thus a "soft" position has a hard time surviving on small grants while a "hard" position merely ends up with few students for a while.