Over the past four hundred years or so, numerous words have been borrowed into English from the languages of the Indian subcontinent. Many of these words were borrowed directly into English by the British colonizers, but some, especially those from Tamil and Malayalam, were borrowed first by the Portuguese, via their colony at Goa, and then almost immediately into English thereafter.
I've compiled a fairly exhaustive list here. There are many other words out there which are claimed to have come from India, but recent etymological research does not bear these claims out. I've double-checked all of these against extremely authoritative sources such as the OED. I've also omitted words which clearly only apply to aspects of Indian culture, such as sari, sati, karma, maharaja, etc., as well as the vast array of Hindu and Buddhist religious terms of Sanskrit origin, leaving only words which have been fully borrowed into English to the point of being largely divorced from their original Indian cultural context.
- avatar - Sanskrit avatara, "descent"
- bandanna - Hindi bandhnu, "tie-dye"
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bandicoot - Telugu pandi-kokku, "pig-rat"
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beriberi - Singhalese beri, "weakness"
- betel - Malayalam vettila, "betel leaf"
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bungalow - Hindi bangla, "Bengal-style"
- cashmere - Kashmir, name of a region in the western Himalayas
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calico - Named after the city of Calcutta
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catamaran - Tamil katta-maram, "tied up wood"
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cheetah - Hindi chita, "leopard"
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chintz - Hindi chint, "chintz"
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chit - Hindi chitthi, "letter, note"
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coir - Malayalam kayar, "cord"
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coolie - Gujarati koli, "laborer"
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copra - Malayalam koppara, "dried coconut"
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corundum - Tamil kurundam, "ruby"
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cot - Hindi khat, "bed"
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cummerbund - Urdu kamar-band, "waist-band"
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curry - Tamil kari, "sauce"
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cushy - Hindi khush, "pleasant"
- dinghy - Hindi dingi, "small boat"
- dungarees - Hindi dungri, "coarse"
- ganja - Hindi ganjha, "of hemp"
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guar - Hindi guar, "guar"
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guru - Hindi guru, "teacher"
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gunny - Hindi goni, "sack"
- jackfruit - Malayalam chakka, "jackfruit"
- jodhpurs, after Jodhpur, a city in Rajasthan
- jungle - Hindi jangal, "wilderness"
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juggernaut - Hindi jugannath, "lord of the world"
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jute - Bengali jhuto, "jute"
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khaki - Urdu khaki, "dusty"
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lac - Hindi lakh, "lac"
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loot - Hindi lut, "loot, booty"
- mogul - Urdu mughal, "Mongol"
- mango - Malayalam manga, "mango"
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mongoose - Marathi mangus, "mongoose"
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pagoda - Tamil pakavata, "devotee of Vishnu"
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pajamas - Urdu pay-jama - "leg-garment"
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pariah - Tamil paraiyar, "hereditary drummer"
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polo - Balti polo, "ball"
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pundit - Hindi pandit, "learned man"
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punch (drink) - Sanskrit panc, "five", an abbreviation of panca-mrta or "five nectars", a combination of five holy medicines
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puttee - Hindi patti, "bandage"
- seersucker - Urdu shir o shakkar, "milk and sugar"
- shampoo - Hindi champo, "to knead, to massage"
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swastika - Sanskrit svastika, "lucky charm"
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teak - Malayalam tekku, "teak"
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thug - Hindi thag, "robber, thief"
- tom-tom - Hindi tam-tam, "drum"
- tourmaline - Singhalese toramalli, "cornelian"
- veranda - Hindi veranda, "balcony"