192.168.1.3 is a private IP address that is sometimes used on local networks. Home networks, particularly those with Linksys broadband routers, commonly use this address together with others in the range starting with 192.168.1.1. A router can assign 192.168.1.3 to any device on its local network automatically, or an administrator can do it manually.
There's nothing special about this IP address versus any other. Your router ether randomly assigns it or it's assigned as a static address, but either way, it offers no performance or security improvements over other addresses like 192.168.1.4, 192.168.1.25, etc.
Automatic Assignment of 192.168.1.3
Computers and other devices that support Dynamic Host Configuration Protocol (DHCP) receive their IP address automatically from a router. The router decides which address to assign from the range it is set up to manage. When the router is set up with a network range between 192.168.1.1 and 192.168.1.255, it takes one address for itself—usually 192.168.1.1—and maintains the rest, or a portion of the remaining addresses, in a pool. Normally the router assigns these pooled addresses in sequential order, starting with 192.168.1.2, then 192.168.1.3, then 192.168.1.4, and so on, although the order is not guaranteed.
Manual Assignment of 192.168.1.3
Computers, game consoles, phones, and most other modern network devices allow manual setting of an IP address. However, simply entering your IP number does not guarantee the device can use it. The network you're on might use 10.x.x.x. addresses, in which case assigning a 192.168.1.3 address simply won't work. The same is true for similar addresses. If your router is dishing out addresses from it's pool of 192.168.2.1 and so on, you can't expect a statically assigned 192.168.1.3 to work.
Problems With 192.168.1.3
Most networks assign private IP addresses dynamically using DHCP. Attempting to assign 192.168.1.3 to a device manually, using a fixed or static address assignment, is also possible but not recommended on home networks because of the risk of an IP address conflict. Many home network routers have 192.168.1.3 in their DHCP pool by default, and they do not check whether that IP address has already been assigned to a client manually before assigning it to a client automatically. In the worst case, two different devices on the network are assigned 192.168.1.3—one manually and the other automatically—resulting in failed connection issues for both devices.
A device with IP address 192.168.1.3 dynamically assigned may be reassigned a different address if it is kept disconnected from the local network for a long enough time period. The length of time, called a lease period in DHCP, varies depending on the network configuration but is often two or three days. Even after the DHCP lease expires, a device is likely to receive the same address the next time it joins the network, unless other devices have also had their leases expire.