Request as a noun means favor;
and as a verb means to ask someone's help..
While require is a verb which means you need someone to do something..
So basically, If you request someone, you're asking for a favor without obliging him to do it so he/she can refuse.It is more like
On the other hand, when you require someone, you're somehow obliging him/her to do it..
Here is an example to make it clearer:
- The company requested the delivery of a sample product in advance for quality assessment.
(Here the company asked for something, that might not be a necessary item or part of the deal ,but it is more of an additional favor.)
- The closing of the transaction required the firm to pay half the amount in advance.
(Here it is an essential part of the deal to pay in advance, without it the transaction is incomplete and might not take place.)