How does gene affect organs development in eukaryotic cells?
The question is very broad and I think can only be answered in very broad terms.

Proteins have metabolic functions. They do stuff! Their actions include catalyzing metabolic reactions, transporting molecules (incl. transport through the membrane), DNA replication, response to external stimuli and many other functions. Humans synthesize almost 30,000 different proteins Muller et al. (2002). The sum of their action put into a specific environment (such as a womb for mammals) end up building up and maintaining an individual.

Let's consider for example Hox genes. From wikipedia 


  Hox genes, a subset of homeotic genes, are a group of related genes that control the body plan of an embryo along the head-tail axis. After the embryonic segments have formed, the Hox proteins determine the type of appendages (e.g. legs, antennae, and wings in fruit flies) or the different types of vertebrae (in humans) that will form on a segment. Hox proteins thus confer segmental identity, but do not form the actual segments themselves.


For more information, I would recommend you to have a look at an intro course to developmental biology. You might want to learn about HOX genes