What is Keigo? And how does it work?

We use honorific speech/keigo to show respect. Being capable of using them properly is essential while talking to your senior or working in Japan.

Today, I'll show you the foundation of keigo as simply as possible, mainly focused on how the three kinds of keigo differ from each other. 

The types of Keigo and how to distinguish them.

Keigo can be divided into three main categories, teineigo, sonkeigo, and kenjougo. Each has different roles regarding how respect is shown, and who's shown respect. 

Seems difficult? No worries, that's why I brought this table.

Polite Keigo: Teineigo (丁寧語)

Teineigo indirectly shows respect to who's listening to you by being courteous. Since it's not like acting modesty or flattery and generally using a few simple words is enough, this honorific is the easiest and most common one of the three types of keigo. You probably should've seen natives using です or ます to conclude a sentence whenever they talk to their elders or in the workplace. 

Also, adding prefixes お and ご to those two suffixes is enough elements to understand the teineigo usages.

Respectful Keigo: Sonkeigo (尊敬語)

Sonkeigo represents respect for the subject of the action. For example, when you say 先生が間もなく教室に来る ("The teacher will come to the classroom soon"), you should use the honorifics to show respect to the teacher. In this case, you can use いらっしゃる (a respectful form of "来る") to do so. Furthermore, if you're talking to a senior classmate, you can add ます (a common suffix of teineigo) and change いらっしゃる to いらっしゃります to show regard for that classmate. 

The confusing part of sonkeigo is that it has the same prefixes お and ご as teineigo, and the way to distinguish them is to pay attention to if the word is tied to a specific person. 
Like, あの方は美しい("That person is beautiful") would turn into あのお方はお美しい with sonkeigo, and 立派な趣味ですね (That's a splendid hobby) would be ご立派なご趣味ですね ("立派" is an adjective verb). Again, these お and ご prefixes are sonkeigo because they're tied to a specific person. Though polite language/teineigo has the same prefixes, they're unrelated to the person. お店 (A shop) and お酒 (Alcohol) are good instances.

In case you're wondering what if "A" in the above table is the same person listening to you, it's okay to fill two boxes with the same person.
Let me show you an example sentence, お酒はお飲みになりますか ("Would you drink any alcohol?"). Here, お酒 is teineigo and お飲みになる is sonkeigo (to insert a verb between お/ご and になる works as sonkeigo). As you can see, sonkeigo and teineigo will not overlap each other even if both of them express respect for the same person.

Note: You can't use any honorifics to respect yourself.

Humble Keigo: Kenjougo (謙譲語)

Kenjougo is used for the object of an action.
Since person A and the speaker is generally the same in the kenjougo circumstance and you can't use it for your action, unlike the other two keigo, kenjougo pays respect by metaphorically lowering your social status, or being modest. 

In this sentence 明日ご自宅を伺います ("I will call at your house tomorrow"), 伺い (a humble form of 行く) is kenjougo because it's your action. Some verbs have each different humble forms like 伺う, but most of the time, you can just put お or ご in front of the verb and する after to make it kenjougo, though there're some alternatives of する.  
Don't worry about the prefixes that came up again, you can easily distinguish them by looking over who's shown respect and how it's shown.

So, that's all for the categories of three keigo. I brought two exercises to distinguish keigo and get used to using the table.

Practice

1) こちらのお品物は袋にお入れいたしますか？ ("Should I put this good in a bag?") お品物, お入れ, いたし, and ます are keigo.

In this case, there're two characters, a clerk and a customer. To fill the table, put the clerk in A and the writer box and the customer in B and the listener box. Since you can't use honorifics for yourself, you can guess this sentence doesn't include sonkeigo. 

From top to bottom, because お品物 is untied to the customer, it's teineigo (after they purchase the good, it will turn into sonkeigo because they possess it and it's no longer unrelated to anyone). For お入れ, this is the clerk's action, so it's kenjougo. Same for the いたし (a humble form of "する"). Lastly, ます is the teineigo suffix. 

2)明日彼にお詫びを申し上げなさい ("Make an apology to him tomorrow) お  詫び and 申し上げ are keigo. 

In this instance, we have three characters, a boss, an employee, and a man. Before filling in the boxes, you need to consider two different times, when the boss is instructing the employee and when they will apologize to the man. So, to make it clear, you can use lower boxes for the former and upper for the latter. Thus, the boss will be in the speaker box, the employee is in the listener and A box, and the man is in the B box.

I'm going to skip お詫び for later since it's slightly complicated. You'll see 申し上げ is the employee's action, so it's kenjougo. 申し上げ is a humble form of 言う.
As for お詫び, you should think this is sonkeigo which is tied to the employee. However, it's teineigo because of the concept of in-group. In-group refers to the structures you belong to, such as your family and the company you're employed in. When you talk to the out-group people, you shouldn't use any honorifics which show respect to your in-group people. For this reason, お詫び is sonkeigo here, but please remember that this case is exceptional.

My opinion on learning Keigo

Keigo is difficult because there're so many words/forms to learn although we barely use them in our daily lives. In other words, mastering keigo would be a good sign that you became as fluent as natives, maybe even more. 
While kanji or idioms-related problems are easily solved by googling, keigo is not. You need to understand what is keigo. If you don't want to be rude, I recommend studying it. 
Although, if you're studying just for chatting with Japanese friends, I would say using です and ます suffixes from teineigo solves like 90% of the problems. They are very useful since they work in a way not being too polite or too casual.

Hopefully, this guide helped you a little about learning keigo. By the way, I rewrote this article which I first made a year and a half back to see how I improved, but guess I still suck. 

Anyways, I hope your effort will bring you a brilliant result!