Politeness

The ~ます form of a verb is the polite present/future tense of a verb.
It has the exact same meaning as the dictionary form, except its more polite. Polite does not mean formal.

The ～ます form should be your baseline when talking to people under normal circumstances.
To give the best English example I can think of:
“Plain” = what’s up.
“polite” = Hello.

Plain form is usually explained as being “friendly”, but that doesn’t properly point out that it’s only “friendly” because its only used with people who are already your friend.
You generally shouldn’t use this with acquaintances, people you do business with, or people who are not “hang out and chat with” friends. 

All this is to say, ~ます form is polite, and the plain forms of verbs are vaguely impolite (but not outright rude)

～ます

～ます form
Ichidan verbs – stem + ます
Godan verbs – stem + います (ignore w)
irregular – する→します　くる→きます。

Ichidan: Verb stem + ます

• たべる→たべます

Godan: Verb stem + います (ignore the w in う ending verbs)

• かえる→かえります
• かう→かいます

Removing ます from this form is called the ます stem of a verb. Note that Godan verbs have and い added. 

ます Conjugations

The ます part itself can conjugate to make other polite forms:

ます Conjugations:
Polite present: ます
Polite negative: ません
Polite past: ました
Polite negative past: ませんでした。

• たべます→たべました　- (polite) I ate.
• かえります→かえりませんでした – (polite) I didn’t go home.
• いきます→いきません – (polite) I won’t/don’t go.

です

です is the ます form (Polite form) of the verb だ。It has its own special conjugations that you need to learn.

です (to be)
Polite present: です
Polite negative: じゃないです
Polite past: でした
Polite negative past: じゃなかったです。

Rule: add です to the じゃない (negative) tenses to make them polite.

• いぬじゃなかったです。ねこでした。- It wasn’t a dog. It was a cat.
• おとこじゃないです。おんなです。(someone is) not a man. (someone is) a woman.
  
  

Regular and Polite Forms Summary Table