detailed explanation

* Quorum : the minimum number of members of an assembly or society that must be present at any of its meetings to make the proceedings of that meeting valid.

For example, if your board has 10 board members, and if it requires a quorum of 5, it means any meeting held with four or less members cannot make any decision.  The meeting with five or more members will be able to vote on motions and make decision. 

* Signing Authority:  who can sign binding documents for the organization, such as contracts, making payments etc. 

Let's say your organization are entering into a contract with the city where the city pay you to care for the homeless.  Your bylaw may require that any contract over $10,000 must have board approval.  The bylaw may require that Board approval means that a majority vote was cast when a quorum is met.  

So if you have 10 board members, and if quorum is 5 members, then this 10,000 contract could be approved by the board when at least 3 members casting yes vote in a 5 member meeting.  

Even when the contract is approved, only member with signing authority may sign the contract.  You may only have 2 members having signing authority (the president and the treasurer).  

On the other hand, the member with signing authority can sign without board approval, when such action does not require board approval.  Things like renting a meeting hall, buying pizza for meeting etc, the member with signing authority can enter into such contracts and spend money from organization.