Yes, a router stops all broadcast packets which are identified with the destination address: FF:FF:FF:FF:FF:FF
Another answer to a similar question answers this in great detail here: How does a router prevent broadcast radiation.
When a router receives a packet, it gets inspected, then forwarded out the appropriate interface or it gets dropped.
When a router receives a broadcast packet, it drops it (excluding directed-broadcasts, dhcp, etc).
When a switch receives a frame, it either forwards it on to a known interface or floods it out all of its ports if it doesn't know where to go. When a broadcast frame comes along, it get's flooded out all interfaces. Every machine in your segment sees it. Excessive amounts of these constitute a storm.
The most common way for a broadcast storm to happen is from a switching loop. If you somehow get a switching loop on your network, these broadcasts will perpetually send this data back and forth forever, or until you remove the loop. This will cause data to hit every machine on your segment. This can cause your network to stop.
When you have a router in between multiple layer 2 segments, each is inherently protected from the other. Remember, a router won't forward on broadcasts.
LAN 1 can be all sorts of messed up, and
LAN 2 will be none the wiser because
ROUTER won't forward
LAN 1 broadcast packets on to anyone.
EDIT: But as mentioned by another user - if the port on your router has been configured to be a switchport, then it will not drop the broadcast packet and instead it will forward it through the switchports on the router.
The router will only drop broadcast packets if the port is configured as a router port.