Suppose you are married and you and your spouse decide to split up. Everything is very amicable and everyone agrees on everything. There really is nothing to fight about but money is (as always) a little tight. You and you spouse both want a separation agreement and a divorce and since everything is agreed decide you will both use one lawyer since that's a bit cheaper.
At first blush it may seem silly but you cannot share a lawyer.
And when you look at the bigger picture that makes very good sense.
The primary duty of a lawyer is to be totally loyal to the interests of the client. As Jesus pointed out you cannot serve two masters – and in the family situation I just mentioned the husband and wife have different interests and they are interests that conflict. It is a good thing that separating couples figure out what they want and agree is fair but as a legal matter if the wife gets more of the family property the husband gets less. And that means if I help my client the wife I hurt my client the husband. As a result a lawyer cannot act for both the husband and the wife.
The idea of conflict is broader than just not acting for two people with opposing interests. If there is anything that could put the lawyer in a position where the lawyer cannot give all their loyalty to the client there is a conflict and the lawyer should not act. So say I want to buy a property in Cambridge Bay. I am bidding on a house there – and a client wants me to help them try to get that very same house. I have a real incentive to make sure my client doesn't get the house – and that means there is a conflict and I should not act. Someone who is a Crown prosecutor cannot properly prosecute a family member or close friend. A plaintiff's lawyer can't legitimately issue a claim against, say, a parent for negligence.
A lawyer had to be outside of the claim and not entangled personally with the situation.
Now sometimes clients can, at least on minor conflict situations, waive the conflict and work with the lawyer anyway . Such situations are rare and only apply where the client has full knowledge of the facts and agrees, anyway, to use the lawyer.