Definition of the concept
The Zero Lower Bound Problem (ZLBP) is a situation in which the central bank of a country wants to lower the short-term nominal interest rates, but faces a hindrance when the interest rate reaches or nears zero, and cannot lower it further. The central bank would prefer to lower the interest rate even more to stabilize the economy, but can’t do so not only because the interest can’t get any lower than zero, but also because of various other reasons.
Firstly, the interest rate cannot go lower than zero and become negative as it would trigger account-holders to withdraw their money from commercial banks and hold it in cash, and consequently, it gives zero nominal return to the banks instead of the negative nominal return they would receive. The account-holders withdraw money from their accounts, because the negative interest which the money incurs deducts funds from their accounts in place of adding to it in the case of a positive interest rate. As a result, the ZLBP limits the central bank to stimulate the market in dire conditions.
Literature on the Zero Lower Bound
The possibility of such an impediment has been mentioned over the years by McCallum (2000) and possibly others. In the recent past, Paul Krugman and Michael Woodford have provided their views about the topic. Paul Krugman stated at the beginning stages of the ZLBP in the US itself that the monetary policy won’t be able to resolve this situation, and unconventional methods will be necessary to tackle it. Speaking on the topic, Michael Woodford remarked that in a zero lower bound, the ideal thing to be done is for the central bank to bridge the output gap by spending liberally on stimulating effects. 
The Zero Lower Bound and the Great Recession
The Great Recession of 2007 brought the ZLBP to a reality when the Federal Reserve sharply lowered the target for the federal funds rate. In December 2008, the federal funds rate was set to the zero lower bound (more precisely in a target range from zero to 25 basis points), and has remained there since then.  Even before the United States, Japan had already faced the pangs of ZLBP in the 1990s, and they could overcome the problem by banking on zero interest rates, and the intervention of the Fiscal Policy. 
ZLBP is a feared phenomenon for the effects that it leaves such as deflation, increase in holdings in cash, decrease in return to commercial banks, and others. The central bank’s involvement, primarily, seeks to attain a targeted inflation rate, and maintain it, but the advent of a ZLBP leads to the opposite as deflation surfaces.
One can also argue that ZLBP is a Liquidity Trap based on the principle that both the concepts revolved around lowered interest rates. However, in a liquidity trap, despite the central banks efforts to lower the interest rates, the interest rates do not show any decrease. While in ZLBP, the intervention from the central bank does bring down the interest rate, and pushes them down to the minimum. Therefore, ZLBP and Liquidity are one and the same is an incorrect argument.
- Theoretical Analysis Regarding a Zero Lower Bound on Nominal Interest Rates by Bennett T. McCallum (2000)
- Simple Analytics of the Government Expenditure Multiplier by Michael Woodford (2010)
- What does Monetary Policy do to Long-Term Interest Rates at the Zero Lower Bound? by Jonathan H. Wright (2012)