I'm sure you're more of a mathematician than me, but here goes.

First, números enteros are definitely integers: negative or positive "whole" numbers or zero (...-3, --2, -1, 0, 1, 2, 3...).
Números reales are all rational or irrational numbers, so they include integers, fractions, decimals, as well as things like π (pi).

So you would think that "división entera" must be "integer division", and I think that in your context it certainly is, but this is not so straightforward, because "división entera" is used in another sense too. In school maths in Spanish you find a distinction between "división exacta" and "división entera", where the first leaves no remainder and the latter does leave a remainder:

"Una división es exacta, cuando al repartir no nos sobra nada, es decir, cuando el resto es 0.
Una división es entera cuando al repartir, nos sobra algo, es decir, cuando el resto es distinto de cero."
http://sanjosedecalasanz5b.blogspot.com.es/2012/12/division-...

This is not the same as integer division, which basically means that you ignore the remainder:

"Integer division is division in which the fractional part (remainder) is discarded"
http://mathworld.wolfram.com/IntegerDivision.html

However, in your case "división entera" does mean integer division. Let me quote the examples for Quick Basic from your own source (whose URL I won't quote, just in case):

"División entera
Print 3\5 ' Resultado= 0
Print 137\5 ' Resultado= 27
Print 30\5 ' Resultado= 6"

Obviously the third is an exact division, but in the first two they've discarded the remainder. So these are integer divisions.

By the way, note the use of the backslash in the examples: this shows it's integer division:

"[Integer division] is sometimes denoted \.
10/3=3+1/3,
so
10\3=3."
http://mathworld.wolfram.com/IntegerDivision.html

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Note added at 2 hrs (2018-05-26 11:00:22 GMT)
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Forgot to add, as a footnote, that "división exacta" in the sense the school blog is using it ("cuando el resto es distinto de cero") is either called division with a remainder or (I think) Euclidean division:

"In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor. Its main property is that the quotient and remainder exist and are unique, under some conditions."
https://en.wikipedia.org/wiki/Euclidean_division

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Note added at 3 hrs (2018-05-26 12:20:29 GMT)
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In my note at the end I meant to say that "división entera", in the sense the school blog is using it, is division with a remainder, not "división exacta" (which has no remainder). Sorry for the confusion; it's bad enough without adding typos as well.

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Note added at 3 hrs (2018-05-26 12:30:32 GMT)
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Ah! OK, but as far as can see "división entera" does mean integer division in computer science.