Cardinalis means "that upon which a door hinges, pivotal" in Latin, from cardo, "hinge, pivot". From this it acquired its secondary meaning, "important, principal", which it still has in English (e.g. a cardinal sin). Its third meaning is derived from this: a cardinal number is a "principal" number, i.e. one that simply says how many objects there are.
That'll be one pound.
Two billion people might die.
Ordinalis means "in order of succession, of an order". It comes from ordo, "order, rank". An ordinal number is an adjective that denotes what place an object has in a certain order. The names of the ordinal numbers are usually derived from the cardinal numbers by adding -th.
That is my second victory.
This is the tenth time she's dumped me.
While a cardinal number refers to several objects ("three apples"), an ordinal number refers to only one of those ("the third apple"). An ordinal number is hence dependent on the notion of a cardinal number: there can't be a third apple unless there are at least three apples. By contrast, there can be three apples without one being the third, if they are just not arranged in any particular order.
The Romans used these terms the same way. They also had distributive numbers, which indicated "every third apple", or apples "in triads", "three each".
These two words have a large semantic overlap, but at the edges there are a few key differences.
Electric is used to describe things pertaining to electricity. It can also be used metaphorically: "the evening was electric".
Electrical can be used nearly everywhere that electric is used when pertaining to electricity (aside from some set phrases). It is not generally used metaphorically in the way electric is. The word electrical can also be used in an additional domain: things concerning electricity. So, generally, people do not say "electric engineer" unless the engineer runs on electricity; instead they say "electrical engineer".
So, in the case of "electric(al) machine" from your question, since you are talking about something that runs on electricity, the two words are essentially identical in meaning.
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There's no difference in meaning between "without xxx" and "with no xxx", but the former is far more common. For example, "walked with no haste" gets only 3 hits in Google Books, whereas "walked without haste" gets 5310 (an extreme example, but the preference is always there).
The other main difference in usage is we tend to avoid "with no xxx" with gerunds (verb +ing). (Note how "with no caring" virtually "flatlines" on this graph).