I'm new to the game and trying to make my first constructed deck. My understanding of the card rule is I can only have four cards of one kind.
What does this precisely mean?
For instance, does this mean I can only have four rat cards total or four rat cards of several kinds? If two different cards do the same thing (like Remove Soul and Essence Scatter), can I have four of each?
Other than basic lands (Island, Forest, Plains, Swamp, Mountain, and Waste) you are only allowed to have four cards in your deck with a given name. (Barring a few exceptions where the card itself says otherwise, or the card is restricted in the format you're playing.)
As an example, you are allowed to have four copies of Plague Rats, and four copies of Ravenous Rats, you can also have any number of Relentless Rats, as the card has specific text allowing "any number" which overrides the four card limit.
In short, you are allowed any number of distinct card names, but only four instances of any given name.
100.2a In constructed play (a way of playing in which each player creates his or her own deck ahead of time), each deck must contain at least sixty cards. A constructed deck can have any number of basic land cards and no more than four of any card with a particular English name other than basic land cards.
You can't have more than 4 copies of one card, except for basic lands and cards that specifically say so in their text box(so far there are only two cards like that - Relentless Rats and Shadowborn Apostle).
Although this rule applies only to Constructed formats. In Limited formats you can include any number of copies of card if you opened them in your cardpool or picked them during draft
100.2b In limited play (a way of playing in which each player gets the same quantity of unopened Magic product such as booster packs and creates his or her own deck using only this product and basic land cards), each deck must contain at least forty cards. A limited deck may contain as many duplicates of a card as are included with the product.
In some formats there are restricted cards of which you can have only one copy in your deck. These cards are listed in official banlists for those formats.
All that matters is the English name of the card, not what it does. You can have at most four copies of a card with a given name:
100.2a In constructed play (a way of playing in which each player creates his or her own deck ahead of time), each deck must contain at least sixty cards. A constructed deck can have any number of basic land cards and no more than four of any card with a particular English name other than basic land cards.
The main exception is the basic land cards: Plains, Island, Swamp, Mountain, Forest, the five corresponding Snow lands, and Wastes. The only other time you can have more than four of a given card is when the card itself says so, like Relentless Rats:
A deck can have any number of cards named Relentless Rats.
So yes, you can have four copies each of several different Rat cards: four copies of Typhoid Rats, four copies of Pack Rat, and so on.
And it also doesn't matter if the cards do the exact same thing, as long as the name is different. You can indeed have four Essence Scatter and four Remove Soul..
Of course, you can only have up to four copies of cards you're actually allowed to play. If a card is banned or otherwise illegal in the format you're playing, you can't have any, and if it's restricted you can have only one copy. See How should I determine whether a deck is legal in a particular format? for discussion of that and other deck restrictions.
Finally, there are also some singleton formats like commander where you do construct your deck ahead of time, but the usual four copies rule is replaced by a one copy rule. Again in those cases, only card name matters, so you can have one copy each of several cards that do the same thing.
All the above are correct.
One note however: in the earliest days of magic this rule did not exist.
Legend has it that this rule-change came to pass after a totally bonkers combo-deck that used a combination of only two cards: black lotus and ancestral recall, both printed in alpha. If you use the ancestral recall correctly, this means you win the game turn one, every single game. People realized this was not fun...
