There are skills, but no skill ranks in 5e
An overarching design principle for 5e (as compared to 3e, 3.5 and 4e) was to simplify the game mechanics and reduce the amount of administration to speed up play. This led to simplifications in several areas including skills.
Skills do exist and are listed page 174 of the Player's Handbook. In 5e, skill checks are just ability checks. For example, sneaking up on a foe might call for a Dexterity (Stealth) check; this is a Dexterity check for which a character's proficiency in the Stealth skill is applied.
Instead of allocating individual skill points to your skills, the system is a simple binary one:1 either you are proficient in a skill, or you are not. If you are proficient, you add your proficiency bonus to the roll. The same bonus is used for all skills (and other checks).
You can try to use any skill, a further simplification is that there is no system differentiating between trained or untrained skills.
So how do I improve my skills?
Instead of manually and individually having to increase your selected skills every time you level by allocating skill points, your proficiency bonus gradually increases as you level, going from the initial value of +2 up to a final value of +6. (This progression is listed in the Character Advancement table at the end of Chapter 1 of the PHB/basic rules, as well as in the class table in the description of each class in Chapter 3.)
You also can improve your ability scores and resulting ability modifiers as you level (using the Ability Score Improvement feature), so you will become better at all ability checks (including those that apply a relevant skill proficiency).
Proficiency bonus does not improve every level, nor do you get ability score improvements every level, so there are levels where your skill bonuses do not change at all. Over the course of your career, your skills will not improve as much numerically as they did in 3rd edition. 5e is designed around a concept called bounded accuracy, which aims to keep the bonuses in a more narrow range.
1 There are also special features for skill-oriented classes like Rogues that give them additional improvements to skills – for example, Expertise allows them to double the proficiency bonus on selected skills – so technically there are more than just two levels of proficiency.