For any skill based ability check the player tells the DM what he wants to try to do, the DM then decides on the appropriate skill or ability and she sets a (usually secret) number, the DC. The player then rolls a 20-sided die to meet or exceed that number, adding his character’s ability based modifier and (if you are proficient in the skill) proficiency based modifier for that skill to the number he rolls.
So you add d20 roll + ability modifier + proficiency modifier (if you are proficient in the skill)
You will usually have the complete modifier for a skill written on your character sheet so that you don't have to think about it too much while playing and just need to adjust it when you reach certain character levels or increase ability scores. So it should really just be a matter of rolling the die, adding a number to it, and telling the DM what you got.
The Number (DC)
Based on how the DM evaluates the difficulty, she sets a number called the DC. Because this is a game where you can do anything and the DM has to adjust things on the fly, this is usually a fairly arbitrary number based on “what feels reasonable” to them, so you just have to trust that your DM is picking a reasonable one as fairly and carefully as they can. The most typical numbers set by most DMs are probably 5, 10, 15, 20, and 25.
Some DMs have more elaborate systems for determining the number. If you are playing a published campaign it is also possible that the authors of that have specified DCs for some situations. There are also times when it will be a “contested roll” against another creature’s skill based ability roll. For perception this is often the case when there is a creature using stealth that you might perceive; your roll would have to beat theirs.
After you succeed or fail on the roll the DM may or may not tell you what the DC was (or whether you succeeded or failed) at her discretion.
Roll and Modifier
The DM calls for a roll and specifies what skill you should be using. Each skill has an associated ability (for perception it is Wisdom) and you will get a modifier from this based on whether your ability score is lower, equal to, or higher than ten. For every 2 below ten you get a -1 modifier and for each 2 above you get a +1 modifier. If you have a 10 there is no modifier. So if you have a Wisdom score of 14, for example you would have a +2 modifier.
Each character also has proficiencies in certain skills based on character creation decisions (background, class, and race). If you are proficient in a skill you get to add your proficiency modifier as well. The proficiency modifier is based on your character level. At level 1 your proficiency is 2, and it increases by 1 every few levels.
So if a level 1 character with a Wisdom of 12 and proficiency in the perception skill made a perception roll they would add 1 to the roll for ability and 2 for skill proficiency to their total score. So if they rolled an 11 it would count as 14. If the DC was 14 or lower they succeed. If it was 15 or higher they fail.
Occasionally the DM will have you make a “straight ability check” when no skill quite fits for what you are doing, in which case you just roll and add the ability modifier without worrying about skills.
This is all much easier once you have your character sheet set up correctly as it will tell you the total to add for each skill.
Various classes have special rules about skill rolls (particularly Rogues and Bards), but you’ll learn those as you go if they apply.
20s and 1s
A very common house rule (possibly the most common of any kind) is that a natural (before adding modifiers) roll of 20 on the die is an automatic success, and a roll of 1 is an automatic failure. At many tables 20 often means a particularly spectacular success and 1 a particularly spectacular failure. But, technically, according to official rules 1s and 20s only have effects like this for attacks and death saving throws.
In some situations the DM will use your “passive perception” for whether you notice something without actively looking for things. This number will equal 10 + the ability and proficiency modifiers. As long as your DM knows what your passive perception is you shouldn’t have to worry about it.