I will have several laptops which will be used by other side to build a local network. Some of these laptops will be working as web servers and other as clients. I will not have any control over the network (I do not know how it will be build and which IP addresses will be assigned to the laptops). So, I wonder how user of the "client" laptops can find available web servers. Can computers in the network be identified by their names. I mean, the IP addresses will be not fixed, but the names of the laptops should be fixed. Can I use names of the computer to identify the web servers. For example on the client side I open browser with the following address in the address line: http://nameofthe_computer.
How do you mean "other side"? do you mean the Russians in the 1950's? Ghosts? the US hockey team?
If it's literally a single network with no routing then there's lots of ways for the machines to find each other based on what OS they're running - for instance simply sharing a resource on most if not all Windows boxes will broadcast their name around the local network, it doesn't have to be the resource you actually want to use, just an empty directory will do. Also there's zeroconf/bonjour, that does something very similar too.
Let us know if it's one big network or if it's segmented in some way and also the OS/s used and we can add some more details ok.
Assuming these are Windows machines and assuming all of the computers will sit on the same subnet (no routing involved), they will attempt to locate the computer by name via a NetBIOS broadcast. This works pretty well on a small network. If you find this causes problems, host files are another option.
Yes, you can, given you have access to the DHCP.
Essentially that is what the internet does. However, configuring your own DNS is quite tricky. If you have access to the DHCP server, you can tell it to also perform simple DNS for you, e.g. using DNSMasq. That is the easiest solution I found, it works in heterogenous environments and allows use of sub-domains (i.e. test1.mylaptop).
Moreover, you can easily integrate that into the internet, e.g. by making test1.mylaptop.office.mycompany.com available.
Without DNSMasq, you'd have to use BIND, but that is very complicated I'm afraid. In that case, I'd stick to the NetBIOS solution.