German language beginners may easily get besonders, besonderes, and Besonderes confused, since they sound nearly the same and have only minor differences in spelling. Once you learn the grammar behind the different spellings, however, you should be able to easily distinguish the reasons for the spelling differences. 

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The adverb besonders (and remember, adverbs modify adjectives as well as verbs), usually translated as "particularly" or "especially," is spelled in lower case (except at the beginning of a sentence) and without the third "e" before the "s."  It will only be used with this spelling as an adverb:
 

So was zum Beispiel läuft besonders gut.

Something like this, for instance, is going particularly well.

Caption 15, Auftrumpfen - Mit Kitsch und Protz

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The adjective besondere (only written with "-s" in its nominative neuter form) is usually translated as "special" or "particular." It is always written in lower case except when it is the first word in a sentence. It will, like all German adjectives, have different endings depending on whether it's in the nominative, dative, accusative, or genitive case, and depending on whether the noun it modifies is feminine, masculine or neuter, and singular or plural.

Here's an example of the adjective besondere in the singular genitive (feminine) case:
 

Einen Wettlauf der besonderen Art liefern sich der

A race of a special kind is being carried out by the 

britische Milliardär Richard Branson

British billionaire Richard Branson

und Hollywood-Regisseur James Cameron.

and Hollywood director James Cameron.

Captions 2-3, Expedition Marianengraben - Zum tiefsten Punkt der Erde

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The word Besonderes, always written capitalized and with the extra "e" before the last "s," is a nominalized adjective, which is an adjective that has been turned into a noun. Note that nominalized adjectives in German are not necessarily so in English! In German, it is usually preceded by either etwas or was (meaning "something" in this context) or the word nichts (meaning "nothing"). 
 

In vielen Familien ist es an Weihnachten Tradition,

In many families, it is a tradition at Christmas

dass es etwas Besonderes zum Essen gibt.

that there is something special to eat.

Captions 3-4, Weihnachtsessen - mit Eva

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Further Learning
To review: besonders is an adverb, besondere (with possible declensions ending with -r, -s, -n, -m) is an adjective, and Besonderes is a nominalized adjective. For more in-depth information on how German turns non-nouns into nouns, read this paper. Then visit Yabla German and search for examples of the above words as spoken in a real world context.