Those are methods for getting a monthly airline schedule.
A "line" is a month's (typically) worth of trips. When someone talks about a "line" they are usually referring to actual trips all month. There are also "lines" of reserve "on call" days but those usually aren't referred to in the same context as "holding a line" - which means you can avoid reserve and actually go fly regularly for the month.
So - line bidding means that someone (union pilots, company workers, a combination) builds schedules using a series of individual trips that have already been constructed before they get put into lines. They do the same thing with schedules of reserve days. This is done for every aircraft type and each seat in that aircraft. Enough schedules (of both flying and reserve) are built so that there are enough for almost every pilot in whatever aircraft and seat they fly. So, the 767 Captains in a particular base can look at their February bid pack and see each individual flying line and reserve line available to them. The #1 seniority pilot picks first and so on. Once the schedules are awarded in seniority order, everyone has their schedule except for maybe 5-20% of the pilots in each fleet/seat (depending on airline). Those who don't are the ones who couldn't hold one of the pre-set schedules or chose not to. Their schedules (typically called secondary lines) will be determined later once the line holders and reserve pilot's schedules have dealt with known conflicts between trips or reserve days and other events like mil leave, recurrent training, vacation and conflicts with trips from the current month carrying over into the new month for which they just bid. The unassigned trips and reserve days that results from those conflicts will be built into new schedules for those 5-20% of the pilots still waiting for their schedules using inputs for what they want (again in seniority order) that they give to the planners.
PBS is essentially the secondary process I just described for the 5-20% applied to the entire pilot group. The flying and reserve days are not built into pre-determined lines (schedules). As a result, the schedules are built to avoid conflicts from the start and there is no need for the secondary process I described above. This is obviously more efficient and requires fewer pilots overall. Instead, everyone inputs their desires for types of trips, days on, days off, reserve if they want it, etc. The schedules are then built using a program that considers seniority, pilot's inputs, FAR legality, contractual rules such as minimum days off, etc. The key driver, as always, is seniority. The number one guy gets pretty much what he asks for as long as it's legal with the FARs and contract. The guys at the bottom get what's left.
You've probably read the pros/cons of each system and the various opinions of each, so I won't go into that again.