Looking at quant roles I keep seeing 'Flow Interest Rates' or 'Interest Rate Flow' - what does this refer to?
The above posts are incorrect in that it has nothing to do with a distinction between prop trading or client facing trading.
Flow trading simply means that the underlying products are "commodity products" in the sense that they are easy to value, high turnover products. Thus rates flow trading involves the trading of cash sovereign government bonds, corporate bonds, overnight index swaps, interest rate swaps, all the way up to caps, floors, and swaptions. Those are rate flow products.
On the other side of the spectrum are "hard-to-value structures" that are not considered flow products, such as exotic derivatives, MBS, ABS, CDOs, CMOs,...
Please note that all products whether flow or not are products traded in client facing roles, thus my first comment that I disagree with Tal's earlier comment.
Floating interest rate is the interest rate that private markets determine, based on demand and supply data. If the demand of a currency goes down, it leads to expensive imports, which leads an extra room for local jobs and services and so on. That is opposed to a fixed interest rate that is set and maintained by a government’s central bank.