Arrays guarantee constant time read and write access, O(1), however many lookup operations (find_min, find_max, find_index) of an instance of an element are linear time, O(n). Arrays are very efficient in most languages, as operations compute the address of an element via a simple formula based on the base address element of the array.
The implementation of arrays differ greatly between languages: some languages allow arrays to be resized automatically, or to even contain elements of differing types (such as Perl). Other languages are very strict and require the type and length information of an array to be known at run time (such as C).
Arrays typically map directly to contiguous storage locations within your computers memory and are therefore the "natural" storage structure for most higher level languages.
Simple linear arrays are the basis for most of the other data structures. Many languages do not allow you to allocate any structure except an array, everything else must be implemented on top of the array. The exception is the linked list, that is typically implemented as individually allocated objects, but it is possible to implement a linked list within an array.