A strum is:
A Strum is the execution of a specific rhythmic pattern, in tempo, in a particular style.
Strumming requires a specific set of skills. They are: 1) Memorization of chords 2) The ability to switch chords smoothly and 3) The ability to choose and execute a suitable rhythmic strum.
Let's break that statement down from the end of the statement to the front.
Each strum is identified with a term that differentiates it from every other strum. This term is typically called a “feel”. Drummers learn these terms early in their studies so learning this language is not only helpful to learning the strums, it’s also helpful with communications among musicians in general and drummers in particular.
A style's term. such as Rock, Country, Samba, FOlk, Bossa Nova, Double Time, Funk, Reggae, Island, etc. don't tell you much about what is actually needed to execute a specific rhythmic pattern. A style's term or name is an organizational tool for later when recalling the elements of a particular style.
Tempo merely is how fast the discernible pulses are going by. It can directly correlate to the downbeats or be implied.
Strumming is inseparably linked to Rhythm.
This starts to get to the heart of what a strum is. It is also often the neglected part of music that is taken for granted. It is something identifiable by non-musicians as something wrong when it is not quite right.
The study of Rhythm is an integral part of really identifying and organizing particular song styles and genres.
One best ways to study rhythm is using Chuck Anderson's Modular Phonetic Rhythm, The Foundation and Workbook 1. I've had great success using this method with students and myself to identify the rhythmic elements of a style.
Once you have started working on mastering the Rhythmic Syllables from Chuck's Modular Phonetic Rhythm, The Foundation and Workbook you can then begin to work on the execution part.
Strumming can be executed with fingers or with a pick or really any device your like. Regardless of your own style of strumming, it ultimately involves combinations of down strokes and up strokes. In the finger style, down strokes can played with the thumb or the nail side of your fingers. Up strokes can be played with the thumb or fingers. Any technique is usable as long as you can differentiate between down and up strokes.
One strum is different from another based on the stroke direction, the stroke density, the subdivision of the beat and the accent pattern.
The tough part is getting your hands to follow your inner, musical ear an execute the desired rhythmic pattern, in tempo for that particular style. Typically the style will play an important role in determining how it is performed.
A strumming patter is NOT, "down down up up down etc..." and not how to organize a rhythmic pattern. On an instrument like the ukulele where gravity effects strumming and a down strum sounds different then the up strum. The verbal commands "down and up" can be helpful to train the strumming hand to follow and convey the pulse of the style and not go off on their own.
Links, Resources on LearningUkulele.com