What is a subsidiary verb? Could someone please explain it and give some examples of its application?
Subsidiary verbs, known as 補助動詞（ほじょどうし） in Japanese grammar, are a small set of verbs which have grammaticalized uses following 〜て. According to Martin†, these verbs include:
In these grammaticalized uses, they have several properties:
They form a single predicate with the verb they follow.
歩いている is usually a single predicate, not "walks and is".
They no longer have their literal meaning (although their grammaticalized meanings are generally related to their literal meanings). The 〜ている ending, for example, expresses aspectual meaning rather than expressing existence.
歩いている is usually the progressive "is walking".
Informally, they often contract with 〜て (see my previous answer for a chart).
歩いている can be contracted informally to 歩いてる.
Because they're used grammatically rather than for their literal meaning, they're generally written in kana:
歩いている should not be written 歩いて居る.
Keep in mind, though, that these verbs aren't always used as subsidiary verbs. When they're used as regular verbs, they can in most cases be written with kanji.
Subsidiary verbs are often referred to as auxiliary verbs, but this can unfortunately cause confusion because the same term is used as a translation of 助動詞（じょどうし）, an unrelated class of inflecting auxiliaries in traditional Japanese grammar including words such as 〜ない and 〜ます.
For this reason, many people choose to avoid confusion and use the unambiguous term subsidiary verbs instead.
† This list from page 512 of Samuel Martin's 1975 Reference Grammar of Japanese. Subsidiary verbs are described in more detail in this book on pages 510-551.