Compare method

Either you implement a compare-method for your object:

- (NSComparisonResult)compare:(Person *)otherObject {
    return [self.birthDate compare:otherObject.birthDate];
}

NSArray *sortedArray = [drinkDetails sortedArrayUsingSelector:@selector(compare:)];

NSSortDescriptor (better)

or usually even better:

NSSortDescriptor *sortDescriptor;
sortDescriptor = [[NSSortDescriptor alloc] initWithKey:@"birthDate"
                                           ascending:YES];
NSArray *sortedArray = [drinkDetails sortedArrayUsingDescriptors:@[sortDescriptor]];

You can easily sort by multiple keys by adding more than one to the array. Using custom comparator-methods is possible as well. Have a look at the documentation.

Blocks (shiny!)

There's also the possibility of sorting with a block since Mac OS X 10.6 and iOS 4:

NSArray *sortedArray;
sortedArray = [drinkDetails sortedArrayUsingComparator:^NSComparisonResult(Person *a, Person *b) {
    return [a.birthDate compare:b.birthDate];
}];

Performance

The -compare: and block-based methods will be quite a bit faster, in general, than using NSSortDescriptor as the latter relies on KVC. The primary advantage of the NSSortDescriptor method is that it provides a way to define your sort order using data, rather than code, which makes it easy to e.g. set things up so users can sort an NSTableView by clicking on the header row.

I would do it as follows:

self.playerItems = [[NSMutableArray alloc] initWithCapacity:11];

NSMutableArray * itemArray;
Item * anItem;
for (int playerIndex = 1; playerIndex <= 10; playerIndex++)
{
    itemArray = [NSMutableArray arrayWithCapacity:60];
    [playerItems addObject:itemArray];

    for (int itemIndex = 1; itemIndex <= 50; itemIndex++)
    {
        anItem = [[Item alloc] init];
        anItem.kind = 1;
        [itemArray addObject:anItem];
        [anItem release];
    }
}

As a side note, you should definitely read up on memory management in Cocoa, since your original code is full of memory leaks. It can be a bit difficult to wrap your head around at first, but once you learn it, it becomes second nature. Well worth the effort.

update:

A more object-oriented way to do this would be to create a Player class, and have each Player manage its own set of items:

Player.h

@interface Player : NSObject
{
    NSMutableArray * items;
}
@property (readonly) NSMutableArray * items;
@end

Player.m

#import "Player.h"
@implementation Player

@synthesize items;

- (id)init
{
    if ((self = [super init]) == nil) { return nil; }

    items = [[NSMutableArray alloc] initWithCapacity:60];
    Item * anItem;
    for (int itemIndex = 1; itemIndex <= 50; itemIndex++)
    {
        anItem = [[Item alloc] init];
        anItem.kind = 1;
        [items addObject:anItem];
        [anItem release];
    }

    return self;
}

- (void)dealloc
{
    [items release];
    [super dealloc];
}

@end

Elsewhere

NSMutableArray * allPlayers = [[NSMutableArray alloc] initWithCapacity:11];

Player * aPlayer;
for (int playerIndex = 1; playerIndex <= 10; playerIndex++)
{
    aPlayer = [[Player alloc] init];
    [allPlayers addObject:aPlayer];
    [aPlayer release];
}

...

[allPlayers release];