You'll find pretty much everything you need in the reference. The tutorials and reference pages are pretty straight forward on how to create all kinds of shapes.

Here's something to get you started. I'll admit it's quite crude, but it demonstrates what you describe above. First I make a rectangle with only three sides, then I use the bezier function to make the fourth side curved.

Code:

size(200,200); //set screen size
fill(255); //make the shape white
beginShape(); //begin building custom shape
vertex(100, 50); //specify vertices of custom shape
vertex(50, 50); 
vertex(50, 100);
vertex(100, 100);
endShape(); //finish building custom shape
bezier(100, 50, 120, 70, 120, 90, 100, 100); //add curved side
 

Well don't throw yourself off too much! It's really quite simple. Since you're drawing a custom shape (i.e. not a plain old rectangle or ellipse), the system needs to know when you're done drawing your custom stuff so it can draw the appropriate lines to connect the dots and to close it off or fill it with a color. By using beginShape() and endShape(), you're simply letting it know the boundaries of your custom shape and that you're done drawing vertices. Using rect() or bezier() does the same exact thing underneath, but Processing hides that stuff from you so you don't have to write out all those points every time you simply want to draw a rectangle. Keep practicing, and it'll all eventually make sense!
 

So it looks like I'm simply creating two filled shapes in the example, no?  So how would I create a filled shape where curved side is an S curve?
Code:

size(200,200); //set screen size
fill(255); //make the shape white
beginShape(); //begin building custom shape
vertex(100, 50); //specify vertices of custom shape
vertex(50, 50); 
vertex(50, 100);
vertex(100, 100);
endShape(); //finish building custom shape
bezier(100, 50, 50, 50, 150, 100, 100, 100); //add S curved side