It depends on the exact constraints your LAN connection places on you.
If you can only connect one physical device to the network (you are only able to get one IP address - probably based on the MAC address of your network adapter), the solution you are looking for is NAT, which creates a private network and gives its nodes some degree of access (and ability to be accessed) from the outside.
The "pure software" solution in Linux would involve:
allowing packet forwarding between different interfaces (which is off by default)
setting up the NAT itself using Netfilter or more precisely its part, the iptables.
configuring routing for the internal network - either statically or via DHCP most likely with the DHCP server running on the machine with LAN connection. This is to ensure that
- the machine connected to the external network knows what to do with packets that are targeted into the private network, and
- the nodes on the internal network know where to send packets for the external network.
This includes setting nameservers correctly (propagating the settings from the LAN, unless you'd want to run your own DNS server - which for one R-Pi is a bit of overkill).
keywords: NAT, iptables, packet forwarding, routing, DHCP, DNS
These first two links I have found (with the first two keywords together) seem to be reasonable.
Of course you can save yourself the trouble of setting this up by buying a home-grade router (starting at roughly a third of the price of a R-Pi). For about the price of the R-Pi, you can get a wireless router.
Now if you are allowed to connect more devices but have only one physical connector, you can set up bridging/forwarding. You would either need to create a bridge containing the two network devices in your PC, or set up packet forwarding and if your LAN uses DHCP (most networks today do) also DHCP relaying. The second option (forwarding) however doesn't make that much sense, since it actually mimics the former (i.e. bridging). There might be some reasons for choosing it, though (for example if you were connected to the network through wi-fi, bridging would be pretty much out of question).
keywords: (network) bridge, packet forwarding, DHCP relay
Again, you can use a hardware switch to the same effect. While these are cheaper than routers, the added value of a router (which can be configured to act just as a switch anyway) makes it usually a better choice.