There are exactly 108 beads in the rosary (japmala). What's the significance of the count 108 here?
Japamala is used for japa (recital) of slokas and/or mantras.
IN Hinduism and other Dharmic religion, there's been tremendous importance given to the figure 108. Many sects in Hinduism requires a person to recite a sloka for 108 times. For eg.
Mukhya Shivaganas are 108 in number and hence all Shaiva sects majorly Lingayats use 108 rudraksha beaded lace for japa. Also they recite supreme lord Shiva's 108 names daily during their morning Shiva-puja.
In Gaudiya Vaishnavism, there are 108 gopis of Vrindavan. Recital of these names, often accompanied by the counting of a 108-beaded mala, is considered sacred and often done during religious ceremonies.
So, mainly it's because of the importance given to the number 108, because of which beads in rosary are 108 in number.
You can read more about the number itself Why number 108 is holy and auspicious in Hinduism.
After a reading this i can't stop myself to post this a beautiful quote
Religion and the arts:The number 108 is considered sacred in many Eastern religions and traditions, such as Hinduism, Buddhism, Jainism, Sikhism and connected yoga and dharma based practices. The individual numbers 1, 0, and 8 represent one thing, nothing, and everything (infinity). 108 represents the ultimate reality of the universe as being (seemingly paradoxically) simultaneously One, emptiness, and infinite.
Hinduism For example: Hindu deities have 108 names, while in Gaudiya Vaishnavism, there are 108 gopis of Vrindavan. Recital of these names, often accompanied by counting of 108-beaded Mala, is considered sacred and often done during religious ceremonies. The recital is called namajapa. Accordingly, a mala usually has beads for 108 repetitions of a mantra.
The distance of Sun from Earth divided by diameter of Sun and distance of Moon from Earth divided by diameter of Moon is approximately equal to 108. It is claimed that the great sires of Vedanta knew this relationship and thus 108 is a very important number in Vedantic chantings.
BuddhismLikewise: Tibetan Buddhist malas or rosaries (Tib. ཕྲེང་བ Wyl. phreng ba, "Trengwa") are usually 108 beads;1 sometimes 111 including the Guru Bead(s), reflecting the words of the Buddha called in Tibetan the Kangyur (Wylie: Bka'-'gyur) in 108 volumes. Zen priests wear juzu (a ring of prayer beads) around their wrists, which consists of 108 beads.
Japa Mala, or Japa beads, made from Tulasi wood. Consisting of 108 beads in total + the head bead.The Lankavatara Sutra has a section where the Bodhisattva Mahamati asks Buddha 108 questions and another section where Buddha lists 108 statements of negation in the form of "A statement concerning X is not statement concerning X". In a footnote, D.T. Suzuki explains that the Sanskrit word translated as "statement" is "pada" which can also mean "foot-step" or "a position." This confusion over the word "pada" explains why some have mistakenly held that the reference to 108 statements in the Lankavatara refer to the 108 steps that many temples have.
In some schools of Buddhism it is believed that there are 108 defilements.In Japan, at the end of the year, a bell is chimed 108 times in Buddhist temples to finish the old year and welcome the new one. Each ring represents one of 108 earthly temptations a person must overcome to achieve nirvana.
108 * diameter of the moon is distance between earth and the moon. And 108 diameters of the sun is the distance between sun and the earth... I don't know if this is true or not but I heard Sadhguru saying this... ~ Video (2.17)