Quality of men's shirts depends on fabric. Formality depends on style.
Basically, there are dress shirts and sports shirts. Dress shirts are worn with suits and ties, sports shirts with sports coats or on their own. Shirts can have different patterns (or no pattern) and different style of collar and cuffs. Cuffs can be buttoned (one or two buttons) or fastened by cuff-links (French style). Collars vary in spread, or angle between edges on the front and width. Sports shirts have collars buttoned down. Men's shirts often have two pleats on the sides of the back, or one square (double) pleat in the middle. A sports shirt can have a small chest pocket. Casual patterned shirts often have two pockets, and sometimes pockets with flaps.
Color and patterns: white and light blue are most common colors for men's shirts. Darker blue, pale pink and yellow, fine strips (most often blue on white), small checkers and pinpoint dots are acceptable in relatively formal settings. Sports shirts can be even plaid and Paisley.
Fabric: usually cotton, sometimes cotton/poly blend, sometimes linen or silk. Sometimes men's shirts are made out of specially treated no-wrinkle (no-iron, non-creasing, etc) cotton. The quality of the shirt depends on type of cotton and thread count. Formal men's shirts are made mostly out of cotton broadcloth or cotton poplin. Broadcloth is fine, tightly woven cloth. Poplin is heavier fabric. Less formal shirts allow for variety of textures and small patterns. They can be made out of Oxford cloth, twill, flannel, or corduroy.
Size: A size of a men's shirt is determined by collar size and an arm length in inches with 1.2" increment. For less formal men's shirts size sometimes boiled down to S (small), M (medium), L (large), and XL (extra large). These sizes vary from brand to brand and are not consistent. A typical American shirt is very roomy in chest and shoulders. My husband's custom-made Oxford shirt from Lands' End had correct neck and arm sizes, but it was so large we could fit in it together. Only it wouldn't fasten at collar, because the neck size was correct. The word for a trimmer, closer fitting men's shirt is "tailored" or "fitted".