From https://learn.adafruit.com/setting-up-a-raspberry-pi-as-a-wifi-access-point/overview
and from http://www.daveconroy.com/using-your-raspberry-pi-as-a-wireless-router-and-web-server/

The following steps are required

Install software

Install hostapd and dhcp

sudo apt-get update
sudo apt-get install -y hostapd isc-dhcp-server

Setup DHCP server

Modify file:

sudo nano /etc/dhcp/dhcpd.conf

change:

option domain-name "example.org";
option domain-name-servers ns1.example.org, ns2.example.org;

to:

#option domain-name "example.org";
#option domain-name-servers ns1.example.org, ns2.example.org;

and change:

# If this DHCP server is the official DHCP server for the local
# network, the authoritative directive should be uncommented.
#authoritative;

to:

# If this DHCP server is the official DHCP server for the local
# network, the authoritative directive should be uncommented.
authoritative;

in the bottom add the following:

subnet 192.168.42.0 netmask 255.255.255.0 {
	range 192.168.42.10 192.168.42.50;
	option broadcast-address 192.168.42.255;
	option routers 192.168.42.1;
	default-lease-time 600;
	max-lease-time 7200;
	option domain-name "local";
	option domain-name-servers 8.8.8.8, 8.8.4.4;
}

Save and exit editor.

Modify file:

sudo nano /etc/default/isc-dhcp-server

change:

INTERFACES=""

to:

INTERFACES="wlan0"

Save and exit editor

Modify file

sudo nano /etc/network/interfaces

after this line:

allow-hotplug wlan0

insert the following:

iface wlan0 inet static
    address 192.168.42.1
    netmask 255.255.255.0

change the following:

iface wlan0 intet manual
wpa-roam /etc/wpa_supplicant/wpa_supplicant.conf
iface default inet dhcp

to:

#iface wlan0 intet manual
#wpa-roam /etc/wpa_supplicant/wpa_supplicant.conf
#iface default inet dhcp

Save and exit editor

To set static ip without reboot:

sudo ifconfig wlan0 192.168.42.1

Setup access point

(create new file):

sudo nano /etc/hostapd/hostapd.conf

insert the following for a wpa-psk secured network (change ssid and passhprase as required):

interface=wlan0
driver=rtl871xdrv
ssid=My_AP
hw_mode=g
channel=6
macaddr_acl=0
auth_algs=1
ignore_broadcast_ssid=0
wpa=2
wpa_passphrase=Raspberry
wpa_key_mgmt=WPA-PSK
wpa_pairwise=TKIP
rsn_pairwise=CCMP

(also change driver is not a rt1871xdrv)

If using an open, unsecure network insert the following:

interface=wlan0
ssid=My_AP
hw_mode=g
channel=6
auth_algs=1
wmm_enabled=0

Save and exit editor

Tell where to find config file:

Modify file:

sudo nano /etc/default/hostapd

change:

#DAEMON_CONF=""

to:

DAEMON_CONF="/etc/hostapd/hostapd.conf"

Save and exit editor

Update hostapd

- in order to support the wifi adapter

get new hostapd:

wget http://adafruit-download.s3.amazonaws.com/adafruit_hostapd_14128.zip

unpak zipped file:

unzip adafruit_hostapd_14128.zip

backup the original hostapd:

sudo mv /usr/sbin/hostapd /usr/sbin/hostapd.ORIG

replace with the new hostapd:

sudo mv hostapd /usr/sbin

change rights:

sudo chmod 755 /usr/sbin/hostapd

Setup NAT

Modify file:

sudo nano /etc/sysctl.conf

aee in the bottom of the file:

net.ipv4.ip_forward=1

Save and exit editor.

To activate without reboot:

sudo sh -c "echo 1 > /proc/sys/net/ipv4/ip_forward"

Create network translation between eht0 and wlan0

sudo iptables -t nat -A POSTROUTING -o eth0 -j MASQUERADE
sudo iptables -A FORWARD -i eth0 -o wlan0 -m state --state RELATED,ESTABLISHED -j ACCEPT
sudo iptables -A FORWARD -i wlan0 -o eth0 -j ACCEPT

to activate on reboot:

sudo sh -c "iptables-save > /etc/iptables.ipv4.nat"

Modify file:

sudo nano /etc/network/interfaces

add to the bottom of the file

up iptables-restore < /etc/iptables.ipv4.nat

Save and exit editor.

Setup daemon to start at boot

register daemon files:

sudo update-rc.d hostapd enable 
sudo update-rc.d isc-dhcp-server enable

OBS!

Depending on the distro it may be required to remove the WPA_supplicant file

sudo mv /usr/share/dbus-1/system-services/fi.epitest.hostap.WPASupplicant.service ~/

Reboot and connect to the access point