I think there a lot of confusion with ARP comes from is in regards to how the IP address and the MAC address work together. The IP address is a layer 3 (network layer) address. The MAC address is a layer 2 (data link) address. The layer 3 address is a logical address. It will pertain to a single protocol (such as IP, IPX, or Appletalk). The layer 2 address is a physical address. It pertains to the actual hardware interface (NIC) in the computer. A computer can have any number of layer 3 addresses but it will only have 1 layer 2 address per LAN interface. At layer 3, the data is addressed to the host that the data is destined for. At layer 2 though, the data is addressed to the next hop. This is handy because you only need to know a host's layer 3 address (which can be found out by using DNS for instance) but you won't need to know the hardware address (and you won't have to bog down the network by sending an ARP request across the internet to find it out). The layer 3 packet (addressed to the destination host) will be encapsulated within a layer 2 frame (addressed to the next hop).
Your computer will have data that it needs to send (I'm assuming that we're using TCP/IP from here on). When the data gets to the Network layer it will put on the destination IP address. All of this info (the network layer datagram, aka packet) is passed down to the data link layer where it is taken and placed within a data link frame. Based on the IP address (and the subnet mask), your computer should be able to figure out if the destination IP is a local IP or not. If the IP is local, your computer will look in it's ARP table (a table where the responses to previous ARP requests are cached) to find the MAC address. If it's not there, then your computer will broadcast an ARP request to find out the MAC address for the destination IP. Since this request is broadcast, all machines on the LAN will receive it and examine the contents. If the IP address in the request is their own, they'll reply. On receiving this information, your computer will update it's ARP table to include the new information and will then send out the frame (addressed with the destination host's MAC address).
If the IP is not local then the gateway (router) will see this (remember, the ARP request is broadcast so all hosts on the LAN will see the request). The router will look in it's routing table and if it has a route to the destination network, then it will reply with it's own MAC address.
This is only the case if your own computer doesn't know anything about the network topology. In most cases, your computer knows the subnet mask and has a default gateway set. Because of this, your own computer can figure out for itself that the packet is not destined for the local network. Instead, your computer will use the MAC address of the default gateway (which it will either have in it's ARP table or have to send out an ARP request for as outlined above). When the default gateway (router) receives the frame it will see that the MAC address matches it's own, so the frame must be for it. The router will un-encapsulate the data link frame and pass the data part up to the network layer. At the network layer, the router will see that the destination IP address (contained in the header of the IP packet) does not match it's own (remember, the IP address has not been touched at all in this process since your computer created the IP packet). The router will realise that this is a packet that is supposed to be routed. The router will look in it's routing table for the closest match to the destination IP in order to figure out which interface to send the packet out on. When a match is found, the router will create a new data link frame addressed to the next hop (and if the router doesn't know the hardware address for the next hop it will request it using the appropriate means for the technology in question). The data portion of this frame will contain the complete IP packet (where the destination IP address remains unchanged) and is sent out the appropriate interface. This process will continue at each router along the way until the information reaches a router connected to the destination network. It will see that the packet is addressed to a host that's on a directly connected network (the closest match you can get for an address, short of the packet being addressed to you). It will send out an ARP request for MAC address of the destination IP (assuming it doesn't already have it in it's table) and then address it to the destination's MAC address.
Now then, I did slightly gloss over 1 part in the above explanation and that's the part about the router finding out the hardware address for the next hop. I just didn't want to disturb the flow with entering into that there. How the router does this will depend on what type of connection (and in some cases, what protocol and/or encapsulation is used on the connection). In some cases, this will be a hard set value (like a frame relay pvc) within the configuration of the router. In some cases, you don't even need a hardware address (like any point to point connection, there's only 1 possible host you could send it to), in those cases the router will just create a data link frame appropriate for the connection and it won't even need to be addressed. This is why the OSI model is good. It's layered so that any layer can change and as long as it takes in information in a standard way (the way the layer above wants to send it) and spits out information in a standard way (the way the layer below wants to receive it), then it's all good. When Frame Relay came along nothing changed with the way you had to address IP packets, all of the changes took place at the data link and physical layers. If some new type of connection comes along in the future, only the data link and physical layers will likely change. When we go to IPv6, only the network layer should change (it probably won't but that's more to do with how the layers tend to blur, but if it were truly layered that would be the case).
Anyways, since I feel like doing an example here's one showing the whole process.
Your computer has an address of 220.127.116.11, it's connected to the 18.104.22.168 network (I'm assuming a subnet mask of 255.255.255.0, we'll call this network 1) which is an ethernet network. Your default gateway is a router (router a) which has an address of 22.214.171.124. That router is connected to the 126.96.36.199 network and the 188.8.131.52 (network 2) network (the interface connected to the 184.108.40.206 network will have an address of 220.127.116.11). The network 2 is also an ethernet network. Also connected to network 2 is another router (router b) which has the address (for the interface connected to network 2 at least) of 18.104.22.168. Router b is also connected to network 3 (22.214.171.124). Router b's interface on network 3 has the address of 126.96.36.199. Here's a (probably bad) ASCII diagram to illustrate:
Network 1 Network 2 Network 3
(188.8.131.52) (184.108.40.206) (220.127.116.11)
Now then, your computer (on network 1 with an address of 18.104.22.168) wants to send some data to a computer on network 3 (with an address of 22.214.171.124). We'll assume that none of the info in already cached in an ARP table on any of the machines or routers (a big assumption but it's mine to make!). Your computer will create an IP packet addressed to 126.96.36.199. That packet will be sent to the data link layer where it needs a MAC address. Based on the subnet mask, your computer will know that the destination computer isn't on the same local network. So, your computer will send out an ARP request for the default gateway's MAC address (ie, what's the MAC for 188.8.131.52?). On receiving the MAC address, your computer will send out the IP packet (still addressed to 184.108.40.206) encapsulated within a data link frame that is addressed to the MAC address of router a's interface on network 1 (because routers have more than 1 interface they can have more than 1 MAC address, in this case each router has 2 ethernet interface each with it's own unique MAC address). Router a will receive this frame and send the data portion up to the network layer (layer 3). At the network layer, router a will see that the packet (which is addressed to 220.127.116.11) is not addressed to router a. Router a will look in it's routing table to find out where to send the packet. The routing table will show that network 3 (the closest match to 18.104.22.168) is reachable via network 2. The routing table will also show the IP address for the next hop is 22.214.171.124. Router a will send out an ARP request onto network 2 asking for router b's MAC address (well at least for the interface connected to network 2). On receiving this, router a will send the IP packet (still addressed to 126.96.36.199, nothing's changed here) encapsulated in a data link frame addressed to router b's MAC address. When router b receives this frame it will do the same thing that router a did, it will send the IP packet up to the network layer and see that the packet is not addressed to router b (the packet is still addressed to 188.8.131.52). Router b will then look up in it's routing table for the closest match and see that it is directly connected to network 3, so there isn't a next hop router to send it to. Router b will send out an ARP request to learn the MAC address for 184.108.40.206. When this is received, router b will send out the IP packet (again, this is still addressed to 220.127.116.11) encapsulated within a data link frame that is addressed to the MAC address of the destination computer. The destination computer will see that the data link frame is addressed to it and will pass the IP packet to the network layer. At the network layer, the IP address will also match that of the computer and the data from the IP packet will be passed up to the transport layer. Each layer will examine the header and determine where to pass it up to until eventually, the data reaches the application running on the destination computer that has been waiting for the data.
What you'll notice through this whole process is that the IP address never changes. The IP packet is always addressed to 18.104.22.168. However, at the data link layer, the address used changes at each hop (it's always addressed to the next hop). As you go up through the layers, you get more and more specific about where the data is supposed to be going. At the data link layer this is very vague, it's basically just, "here's who to pass it on to, they should know what to do with it". At the network layer you get more specific (this is the exact computer I want to send this to). Above that you get more specific (is it TCP or UDP?, what port?, etc)
please see the below animated link for more understanding of TCP/IP at layer 2 or 3 level.