In India, it is tradition of touching feet of a relative who is elder in age (like grandfather/mother, uncle/aunt, in-laws) for blessing.
What is the reason/story behind this ?
On the human body, one's head is considered the most honorable part, where your knowledge, mind, consciousness reside, and the feet are considered least honorable, as they are closest to the ground, dirt, elements.
Touching an elder's feet is a huge sign of respect, you're effectively communicating that, even their most dishonorable part is respected by you and is higher than your most honorable part. It's a sort of bowing down to the experience and wisdom of the other person. The elder normally will respond by placing a hand on your head, acknowledging your most honorable part. Sometimes they'll also verbally give you a blessing, "be happy always" or "may God protect you" something like that. This blessing is called "ashirvaad" and obtaining this from an elder is special.
The touching of the feet in Indian culture, is to show of respect and it is often an integral part of darshan, or "visions of the divine" in sanatan dharma worship. When greeting, children touch the feet of their family elders, while people of all ages will bend to touch the feet of a great guru, deity of a Deva (God) (such as Rama and Krishna).
Source : Wikipedia
So the main reason for touching feet is to show respect to elders and get their blessings. It is also said when you touch feet of elders you inherit their conduct(आचरण in Hindi).
Sid M has explained well, I agree with him, but I want to explain more.
There is always energy of one's soul in the body. Most of people can increase this energy by virginity, meditation or yoga or any other way (If exists I do not know).
Now this energy can be transferred to other soul by 4 ways:
1.> By eyes
2.> By toes
3.> By hands
Most Hindu God's statue showing hand
Saints put hand on head
4.> By wishing/blessing from heart
When people touch feet of saints/elders/teachers, they do not actually bow to their body, but his pure life or pure soul. And they wish to have their goodness as blessings from them.
Yes Sid M is absolutely right that for showing blessings and respect we touch feet of elders.
Taking darshan by touching the feet is an old vedic tradition. The Laws of Manu say:
II. 71. At the beginning and at the end of (a lesson in the) Veda he must always clasp both the feet of his teacher, (and)...
II. 72. With crossed hands he must clasp (the feet) of the teacher, and touch the left (foot) with his left (hand), the right (foot) with his right (hand).
It was over time extended to others one should consider as superiors as well.
Before embarking on any auspicious occasion seek the blessings of elders. Always when you bow and seek blessings they wish you all the good luck (Sukhee Bhava, Sri rasthu, Subha masthu, Dheergayu rasthu), etc.
In fact at the beginning of every endeavour seeking the blessings of Mother is most venerable, next you should think of your family deity (kula daivam) then your grama devata (the lead god of your town).
All traditions or parampara trace their origin back to Bhagavan - Origin of Rishis. Human tradition in Sanatana Dharma trace their origins to Rishis and who in turn were taught by the primordial Guru.
So why are the feet of Bhagavan so important ? Trivikrama incarnation might give a clue. Touching the feet of elders is symbolic of touching Bhagavan's feet which represent what the Bhagavatam expound.
Elders represent the Rishis and Rishis represent Bhagavan. So by touching the feet of elders at appropriate times the link to Bhagavan is maintained. Simultaneously the import of what the feet represent from Bhagavan's perspective represent in a deeper sense must be kept in the mind of the saadhaka. It is not a mere mechanical gesture of touching an elder's feet. Bhagavan's usage of the feet to remove impurities in the material universe must be kept in mind.