The subnet mask identifies what portion of the IP address is used to identify the network and what portion identifies the host. For a complex discussion, everywhere the subnetmask is a 1 (when written in binary) the IP address would identify the network and everywhere the subnetmask is a 0 the IP address would identify the host. In a complex envionment the subnet mask can be variable for different subnets (as long as the routing protocol for the WAN supports it) and need not use just 255 and 0. For example for point to point circuits where you would only ever have 2 hosts I use the subnet mask of 255.255.255.252. So much for that.....
To make it easy: where the Netmask is 255 the IP address is the network and where it is 0 the IP address identifies the host. Most simple networks use this numbering scheme.
To configure a network properly all the hosts on a LAN use the same network mask and the same network address (caveot: some complex configurations can be set up differently but it is generally not recommended).
In your case with a netmask of 255.255.255.0, the network address is 192.168.1.0 and the host addresses are 1 and 10.
More complex information:
RFC 1597 identifies the address space that is to be used for private networks so that networks can be set up without creating a black hole.
Imagine you used the network 204.71.200.0 netmask 255.255.255.0. You computer could not get to
www.yahoo.com 204.71.200.67 because it would think it was local - thus the black hole.
The addresses that will never be issued as a public Internet address are:
10.0.0.0 - 10.255.255.255
172.16.0.0 - 172.31.255.255
192.168.0.0 - 192.168.255.255
You can choose a subnet mask however you wish as long as your network address falls into these ranges. Note that the reason there are so many ranges is that the subnet mask used to be a default based on the network address (Class A,B,C networks). As the Internet grew it became necessary to allow users the ability to define their own subnet mask.
Personally I always use 10.x.x.x for private networks.
More than you wanted to know?