This is called finding the k-th order statistic. There's a very simple randomized algorithm (called quickselect) taking O(n) average time, and a pretty complicated non-randomized algorithm taking O(n) worst case time.
Everything you need is in these powerpoint slides. Just to extract the basic algorithm of the O(n) worst-case algorithm:
Select(A,n,i):
Divide input into ⌈n/5⌉ groups of size 5.
/* Partition on median-of-medians */
medians = array of each group’s median.
pivot = Select(medians, ⌈n/5⌉, ⌈n/10⌉)
Left Array L and Right Array G = partition(A, pivot)
/* Find ith element in L, pivot, or G */
k = |L| + 1
If i = k, return pivot
If i < k, return Select(L, k-1, i)
If i > k, return Select(G, n-k, i-k)