A line (or point-to-point) network is where the computers are simply connected to each other in a line. Messages are then passed along the line, through each node, until they reach their destination. This means that messages are easy to intercept, as if the message passes through a node, the node can take a copy. It also relies on all the nodes being switched on - if a node was turned off or developed a fault then the network would be "split" at that node, as communications cannot pass through it. Finally, it relies on every node having two NICs, which is an additional cost, as most computers only have one NIC integrated into the motherboard.
A bus network is very similar to a line network, except each computer has an individual connection to a shared line When messages are transmitted along to shared line, all nodes receive the message, In theory, all the nodes except the node the message is addressed to will ignore the message. However, as with the line network, a malicious node could intercept communications, so bus networks aren't very secure.
The main advantage of a bus network is that it is cheap to set up - the only hardware that is needed is a shared line, a line between each node and the shared line, and one NIC per node. However, this is offset by the fact that if there is a fault on the shared line then the entire network could grind to a halt, in the same way as a line network.
A tree network has one "root", with several communications lines coming off it, which may then "branch" themselves. The weakness with a tree network is that if the root develops a fault, then the branches cannot communicate with each other (although communications should still be successful within the branch).
A star network has one central server, and several nodes running off it. If any one of the nodes fails, the network will still operate (although communications to the faulty node will fail). However, if the central server fails, then the entire network will fail. Star networks also offer high performance (assuming the central server can cope with the number of connected nodes), as each communications line is dedicated to a single node.
A ring network is like a line network, but the two ends are connected. Data only travels in one direction (clockwise or anti-clockwise) to minimise packed collisions, although this means large ring networks can be very slow.
A mesh network is where nodes are connected to each other, but some nodes may be connected to more than one other node. This leads to high levels of redundancy, so if one communications line or one node develops a fault, the network will still operate, although it may slow down. For similar reasons mesh networks are very fast - if one communications line is congested, then an alternative path is used. The internet is connected in a mesh-style network.
A fully connected network is very similar to a mesh network. It is where every node is connected to every other node. Because of this, although it is very fast, it is highly impractical for networks larger than a few nodes, as each node needs enough NICs to connect it to every other node.