Correlatives (Korelativoj or Tabelvortoj) is a system of 45 words, partly pronouns (kiu – who, tiu – this, kies – whose, etc), partly adverbs (kie – where, tie – there, kiom – how much, etc).^[7]

Every correlative consists of two parts – first and second.
First parts^[8]:
ki- = interrogative (demandovorto)
ti- = demonstrative (montrovorto)
i- = indefinite (nedifinita vorto)
ĉi- = universal (tutampleksa vorto)
neni- = negation (nea vorto)
Second parts:^[9]
-u = individuality (individuo)
-o^[10] = thing (aĵo)
-a = quality (kvalito, eco)
-es = possessor (posedo)
-e = place (loko)
-am = time (tempo)
-al = cause (kaŭzo)
-el = manner (maniero)
-om = quantity (kvanto)
By combining these two sets, it is possible to form 45 words:

Correlatives of individuality (-iu) can form accusative and plural.
Correlatives of things (-io) can form accusative, but normally do not form plural.
Kion tiuj homoj ĉion ne elpensas. – What all do the people think out!
Adverbial correlatives of place (-ie) can form accusative to mark direction.
Mi estas tie. – I am there.
Mi iras tien. – I am going there.
Other adverbial correlatives (time – -am, cause – -al, manner – -el, quantity – -om) and possessive correlatives (-ies) do not decline.

It is not normal to use first or second part of correlative and combine it alone with some root or affix. However there are some exceptions – neni-aĵo, neni-eco, neni-igi, neni-iĝi, ti-aĵo.^[11]
Neniaĵo – nearly nothing, thing with no value
Viaj kontraŭuloj fariĝos neniaĵo kaj pereos.^[12] – Your enemies will do nearly nothing to themselves and they will perish.
Nenieco – quality to be like nothing, nothingness
dezerta regno de la nenieco^[13] – desert kingdom of nothingness
Neniigi – destroy
Mi neniigos vin, kaj vi ne plu ekzistos^[14] – I will destroy you and you will no more exist.
neniiĝi – disappear
Li disneniiĝis kiel fumo. – He disappeared like a puff of smoke.
tiaĵo – thing of that quality (ti-aĵo, from tia and aĵo)
Mi ne ŝatas tiaĵojn. – I don’t like things that look, behave, etc like that.

Some correlatives can accept different category endings, some can accept suffixes and some can even form composites with other roots. Very often is the set of possible derived forms restricted to some few traditional forms. I will go through one type of correlatives after another.

Individual – -iu
The individual form can precede virtually any root (see –ia):
tiumomente – in that moment, tiunokte – during that night, tiusence – in that sense, kiusence – in what sense, iusence – in some sense, etc.
It is impossible to add any ending or true suffix to it. Of course, it can be declined.

Quality – -ia
The correlatives of quality can be also before nearly any root. The difference usage of -iu and –ia, is implied by their meaning – -iu refers to some concrete thing, occasion, etc., -ia refers to some type, quality or style of thing, occasion, manner etc. Sometimes it is hard to distinguish these two groups.
tiamaniere – in such manner, tiasence – in such sense, tiaspeca – of such type, similar, analogous, kiamaniere – in what manner, how, ĉiamaniere – in all manner

Place – -ie
It is possible to form adjectives from the correlatives of place (-e), e.g. tiea, ĉiea, etc., with the meaning “finding itself there, everywhere, etc.”^[15]. These adjectives are normally declined.
La ĉiea pluvo detruis ĉiuj vojojn. – Rain that was everywhere destroyed all roads.
Sometimes, it is also possible to see form tieulo – the man from there.

Quantity – -iom
The ordinal form of the correlatives of quantity is created by adding the ending a, e.g.: kioma, tioma.
Sur la kioma etaĝo vi loĝas? – On which floor do you live?
It is also possible to diminish or augment the quantity (practically only iom): iomete – a bit of, iomege – some large quantity, iometo – a bit. Some of the numerical suffixes can also be added: kiomoble – how many times, kiomfoje – how often
The forms with adverbial e are only emphasized forms of the original: iome.

Time – -iam
The forms with adjectival ending (-a), e.g. tiama, ĉiama. iama, etc., with meaning “existing in that time, existing always, existing in some time (in the past).

Cause – -ial
The only derived word is kialo – the reason, motive.

Manner – -iel
Tiele and iele are emphasized forms of tiel and iel.
The forms with adjectival ending (-a), e.g. tiela, kiela etc. are equivalents to tiamaniera, kiamaniera, etc.
It is also possible to see word *tielmaniere – the official form is tiamaniere or tiumaniere.

Possession – -ies, Thing – -io
I do not know about any derived forms.

^[7] In other languages is very often also some system of some adverbs or pronouns, but mostly it is not so regular and complex as in Esperanto. See for example English (Source: J.M.D. Meiklejohn, The English Language - Its grammar, history and literature, 1895):

^[8] There are also unofficial forms with the first part al – another (aliu – somebody else, aliel – in another way, etc.) This set was created by analogy from the word alia – another. The words derived from the official root and the unofficial set of correlatives have different meaning.

^[9] Some of the second parts are same as normal endings, but they have different meaning – ordinary u stands for volitive, ordinary e stands for any adverb, not only for place, ordinary a stands for any adjective, not only for quality. Only o has nearly the same meaning.

^[10] This o has nothing to do with the noun ending o, so it is impossible to replace it by apostrophe.
^[11] PMEG – http://purl.oclc.org/NET/pmeg/pmeg10/taqord.htm – down
^[12] Cited from PMEG, originating from Old Testament translated by L. Zamenhof

^[13] Cited from PMEG, originating from Schiller, F.: La Rabistoj, translated by L. Zamenhof

^[14] Cited from PMEG, originating from Old Testament translated by L. Zamenhof
^[15] PMEG – http://purl.oclc.org/NET/pmeg/pmeg10/el_e.htm, subchapter Vortfarado