As you know, Russian adjectives can have long and short forms. Nevertheless, firstly, we can "shorten" not all adjectives. And secondly, long and short adjectives not always replace each other and the usage of short form always conveys certain meaning. One more thing that you should remember: we don't decline short adjectives in cases.
Этот мужчина сильный (long form) - This man is strong.
Этот мужчина силен (short form) - This man is strong.
There are three ways to make the short form of adjectives.
1. Remove the ending of a long adjective, thus we get the short form - for masculine gender, add the ending "-а" [a] - for feminine gender, "-о" [o] - for neuter one or "-ы" [eh] - for plural form.
Сильный [sil'nyy]: силен [sil'yon], сильна [sil'na], сильно [sil'na], сильны [sil'ny] - strong.
2. Add the extra letter "o" [o] in words with combination of a consonant and the letter "к" [k] (between them). Otherwise, the rules are the same.
Сладкий [slatkiy]: сладок [sladak], сладка [slatka], сладко [slatka], сладки [slatki] - sweet.
3. Add the extra letter "e" [e] in words with combination of a consonant and the letter "н" [n] (between them). Otherwise, the rules are the same.
Бедный [b'ednyy]: беден [b'ed'en], бедна [b'edna], бедно [b'edna], бедны [b'edny] - poor.
But how should we define, whether we can make the short form and have such word in Russian? The answer is simple: we can make the short form only from those adjectives which can form the comparative degree. If we can use a long form with the words "более" [boli'ye] (more) and "менее" [meni'ye] (less), then it's possible to make the short form.
Более сильный [boliye sil'nyy] - stronger, менее сильный [meniye sil'nyy] - weaker.
Силен [sil'yon], сильна [sil'na], сильно [sil'na].
Более старый [boliye staryy] - older, менее старый [m'eniye staryy] - younger.
Стар [star], стара [stara], старо [staro].
We can use the short form in a sentence, when there is no noun after a long adjective.
Мой брат удачливый [Moy brat udachlivyy] - My brother is lucky.
Мой брат удачлив [Moy brat udachlif] - My brother is lucky.
We often use the short form, when we need to describe human condition at the moment.
Он весел [On ves'el] - He is funny.
Он богат [On bagat] - He is rich.
Он болен [On bolin] - He is ill.
Attention! Don't mix up short adjectives with adverbs. They can be written in the same manner but always sound in different ways. Pay your attention to the rest of words in a sentence. If a noun being in combination with adjective is in any case except of instrumental one (with the verb "быть" [byt'] (to be)), it's impossible to use the short form.
Я видел красивую собаку - I have seen a beautiful dog.
(We can't use the short form in this case).
Мой начальник был строгим - My boss was strict.
Мой начальник был строг - My boss was strict.
(In this case it's possible to use the short form).