Siddha has no equivalent in english language, it can be understood as combination of perfection and establishment(when both are used simultaneously i.e. Perfection+Established or perfectly established).
In hanuman chalisa the siddhi word means that person has attained such qualities by establishing himself into perfect self or atman.
In Hinduism, eight siddhis (Ashta Siddhi) or Eight great perfections (mahasiddhi) are described as follows
1) अणिमा Aṇimā: reducing one's body even to the size of an atom
2) महिमा Mahima: expanding one's body to an infinitely large size
3) गरिमा Garima: becoming infinitely heavy
4) लघिमा Laghima: becoming almost weightless
5) प्र्राप्ति Prāpti: having unrestricted access to all place
6) प्राकाम्या Prākāmya: realizing whatever one desires
7) ईशत्व Iṣṭva: possessing absolute lordship
8) वस्तव Vaśtva: the power to subjugate all
Source Shrimad Bhagwat MahaPurana:
The masters of yoga speak of eighteen mystical perfections [siddhis] and meditations [leading to siddhis]. Eight of them are [primary and] prominent in Me and ten of them [are secondary and] originate from the quality [of goodness]. Oh! gentle one, the eight mystical perfections consist of the
ability to get, as for the form, into the smallest [animâ],the biggest [mahimâ] or the lightest [laghimâ relative to garimâ, the heaviest], the ability to acquire whatever material object [prâpti], the ability to enjoy sensually whatever can be seen or heard [prâkâmya], the ability to have the upper hand in employing the forces [îs'itâ or îs'itvâ], the ability to be in control by means of magic unobstructed by the modes [vas'itvâ] and the ability to answer to any desire that seeks [His] favor [kâmâvasâyitâ].
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Nidhi
In Sanskrit Nidhi means treasure there are 9 (primary)types of treasures or valuable possessions.
The nine nidi's are viz
1) mahapadma "great lotus flower"
2) padma "lotus flower"
3) shankha "conch"
4) makara "crocodile"
5) kachchhapa "tortoise"
6) kumud "a particular precious stone"
7) kunda "jasmine"
8) nila "sapphire"
9) kharva "dwarf"
Sources:
Primary
Secondary
More on nava nidhis
Possession of Siddhis and Nidhi's:
Siddhi's can be acquired by anybody in this universe who has enough Sadhana Balam or Tapobalam (certain spiritual height/depth of realization of practitioner), here is a way to posses mantra siddhi also if you are a bramhin(by karma) you can posses gyatri siddhi by doing n number of mahapurascharana's, but it is just a waste of time if you want to get out of the cycle of life or samsara see this answer for more on ultimate goals in hinduism.
There were many siddhas in past who possesed such abilities, they still exist and they are innumerable.
Nidhi's are primarily possessed by Kuber because he is prime minister of Goddess Laxmi (Goddess of wealth).
