Abstract ideas are universals - it is also referred to as concepts. A universal is a quality; a particular is a bundle of qualities.The best way to single out a quality is by pointing out what different particulars have in common - this process is called abstraction. All concepts are not equally abstract: the following examples show that there are hierarchies of concepts and that some concepts are extracted from other concepts.
Dog, red and person are not concrete; love, hate or wisdom are not as abstract as one thought.
Red is a class of sensations. No one has ever seen red alone; no one can imagine red independent of other visual properties such as area, shape, depth, brightness, etc, although it is not difficult to imagine a red thing, e.g., Clifford the Big Red Dog. Red is actually abstract - hard to believe, I know - it is the name of a universal. Names standing for particulars are called proper names, e.g., Clifford the Big Red Dog; a name that stands for nothing is a nonsense - from here we depart from the Platonist view. Whether the Platonist view is true or not I personally do not think it's worth considering at all.
Dog stands for a class of animals each of which we call a dog; no one can see or even imagine what dog looks like, although one can easily imagine an instance of dog - Clifford the Big Red Dog, for example. Dog is actually a universal or an "abstract idea" or a concept. On this account, the English language, which puts an indefinite article before a noun, cultivates a very good habit of mind: an indefinite article before a noun indicates that the noun stands for a universal - people have been thinking like this all the time, but it takes a Bertrand Russell to point it out. (English language is a manifestation of a superior mind: there is a famous Chinese paradox "white horse not horse," which is no paradox at all in English, owing to the English indefinite article; this example illustrates how language affects thinking.)
In Mauro Allegranza's answer, Fido, a proper name in the strictest sense, is used to stand for a particular. This is the kind of precision we should at least make an effort to practice in here.
In the case of a particular dog, everything a person knows about it he knows through his sensations in his own head. The dog in itself as an individual is not the origin of our knowledge about it; it is the sensations in our own head that are the origins of our knowledge about the dog. We believe the dog was the beginning of a chain of physical events that lead to our sensations, but everything we can say about the physical world is inferred from sensations; as a matter of fact, the causal chain between the dog and the canine patch of colour in the mind is open to doubt: there is no evidence to suppose we are not dreaming. It follows that in order for a proper name to have meanings, it is only necessary that it invokes mental images; this explains why Hamlet is considered a fiction while Louis XIX is considered meaningless: there is a whole play describing Hamlet in picturesque terms, whereas Louis XIX invokes no mental image.*
Like dog and red, love is also a universal or an abstract idea, but it is not detached from particulars: it is a class of particular mental states or feelings. An instance of love is not intrinsically different from an instance of seeing a dog: they are all mental occurrences, although we believe they both have physical causes.**
All universals derive their meanings ultimately from particulars: some are classes of particulars, some are class of classes of particulars, some others are class of classes of classes of particulars, etc; there are hierarchies of concepts - that is why Bertrand Russell said he discovered the theory of types because that is how people actually think. For example:
Red is the class of all red sensations; red is a colour, but "Clifford the Big Red Dog is a colour" is not true, therefore colour is a class of classes.
2 is the class of classes each of which has only two members; number is a class of numbers and is therefore a class of classes of classes.
A cleat is a member of a carefully selected pair; a pair of cleats is an instance of 2; 2 is an instance of number.
Regarding everything that exists only in thoughts as abstract ideas - many dictionaries are still defining abstract this way - has licensed many nonsenses disguised as "abstract concepts" and has given aura to a mysterious manner of speech that sounds like announcing aphorisms containing profound truths but is actually emitting nonsenses. If a word stands for neither a universal nor a particular, it is a nonsense, or an incomplete symbol at best.
- Please do not mistake me for an idealist. Idealists believe the existence of things depends on the mind; what I'm talking about here is our knowledge: our knowledge of the external world - including our knowledge of the existence of things - begins from the mind, not the external things. If you are inside a submarine, everything you can say about the outside depends on what the sensors tell you. Knowing the true origins of our knowledge does not increase the credibility of the mind; on the contrary, it gives us more reason to doubt everything we firmly believe.
** Germans don't say "Ich liebe dich" very often because when they say it they mean it. Americans say "I love you" all the time as a compliment - whether or not this expresses a mental state does not matter - but this kind of lies are fully approved of by Shakespeare.
Source: Russell, Bertrand. "Knowledge by acquaintance And Knowledge by Description." The Problem of Philosophy.