Lucian is a new Marksman, and plays a bit like an ranged AD caster a la Graves or Miss Fortune. While not a hard carry in the traditional sense, he does bring some of the same tools to the table. Let's have a look at him in detail.
He has a 550 range and gets 3.3% per level attack speed scaling and 3.0 AD scaling. Those are all good ADC stats in general. Compare a couple other champions
Ashe: 600 range, +4% attack speed, +2.85 damage
Graves: 525 range, 2.9% attack speed, 3.1 damage
He's a bit on the squishy side for health (390 + 80 per level) but has a slightly elevated armor (15). All in all, a fairly middle to slightly below average ADC just looking at his stats.
For a lot of champions, passives, along with ults, give most of the flavor to the champion in question. In the case of Lucian, it also gives him most of his power. Lightslinger lets him double attack within 6 seconds of using an ability. It deals 50-75% of his AD (scaling up with level), but also applies on-hit affects. The second shot can critically strike.
This is an incredible tool for a variety of purposes. It makes life steal very valuable on him. Blade of the Ruined King looks to be an exceptional early and late game item, letting him potentially shred health better than any other ADC except Vayne. Because the shots can crit, the passive scales well with things like IE and even Zephyr - which offers CDR and crit chance in a nice little package.
Piercing Light (Q)
The range on Lucian's Q is the same as his AA range (550) and it does 80-240 physical damage + 0.6-1.2 AD. Note the scaling AD percent (which is unusual). It's a significant incentive to level this, especially if you built Bloodthirster over Ruined King first.
The ability itself is a relatively quick beam of light that is roughly a minion long. There's a slight delay making it just barely dodgeable. It has a pretty short cooldown (dropping from 9 to 5 seconds by end game).
Note, the range on this one is weird. It MUST target a champ but doesn't necessarily hit the champ (think Jayce's leap attack). It can only target someone within his AA range (550), but then travels 1100 total.
Ardent Blaze (W)
With a much longer range (1000) and the ability to temporarily grant vision, this is Lucian's "brush check" skill. It deals magic damage with a 0.9 AP and 0.6 AD ratio built in. Targets hit by it, or by its cross shaped explosion, are marked, and follow-up attacks from Lucian on marked targets give him +40 move speed.
It is not super impressive, and unlikely to be a big source of damage from Lucian. Use it to brush check and to proc your passive and/or your Spellblade on Trinity Force if you go that route. It does mean that Lucian will frequently be faster than the people he is chasing, and his positioning advantage will be even greater in combat.
One trick people have already noticed is that statuses like Spirit of the Elder Lizard and Red buff will continuously proc the speed buff.
Relentless Pursuit (E)
If you're like me, the first thing you noticed was that Relentless Pursuit - an otherwise simple dash forward - has a very long range (425) and a down scaling mana cost that ends at 0 mana at level 5. That's a first, that I'm aware of, and means that the skill functions with significant differences late game. It allows for pretty impressive wall jumps (Ezreal's range on his E is 475 for comparison). I haven't had a chance to test him extensively to know whether it's a "flash" (like Ezreal) or a "jump" (like Tristana), so the interaction with Nautilus/Blitz/Thresh hooks is unclear (please post below if you've figured it out).
Relentless Pursuit also has a kill reset component, but only if the kill happens while his ultimate is active (more on that in a moment). It also removes slows, meaning that, again, Lucian is going to be really hard to catch or escape from. Please note, it removes all move speed debuffs. That means exhaust falls off (or at least appears to). It removes Wither and Ice Blast, making him a fantastic counter to Nasus and Nunu. If you're laning against Lucian as a support, don't take Exhaust. Just go with Ignite.
The Culling (R)
Probably his most controversial ability. The Culling gives Lucian rapid fire shots for 3 seconds with the number of shots scaling with his attack speed. It's a channel and can be interrupted (unlike a Fiddle ult, for example), but he can move and use Relentless Pursuit during it. The damage per shot is not a normal attack, but is 40-60 + 0.25 AD + 0.10 AP. He'll fire anywhere from 17 to 34 shots depending on his attack speed at the time of execution.
The ability locks Lucian into a direction (he can cancel the ult early to stop the locked direction). The culling also does quadruple damage to minions (NOT monsters) and can be used without getting on the other side of a minion wave for that reason.
During development, the AP ratio was higher and people were routinely getting off 10k damage attacks with full AP builds (go watch Youtube videos, they're hilarious). Now, the max damage at lvl 18 is 2040 + 8.5*AD + 3.4*AP which means somewhere around 5k total damage if you land everything (recognizing that it's realistically impossible to max both attack speed and achieve a ~400 AD).
The shots fired from the culling do not proc on-hit affects nor do they proc life steal. They do proc spell vamp, but at a third of the rate. Still, this might make Will of the Ancients a viable item for a support to build in the right team comp. They also don't count as single target for things like spellvamp and Manamune (no abusive Manamune procs here).
Lucian's abilities scale reasonably well off of AD giving him a likely build path similar to Graves and MF. Bloodthirster into IE with Black Cleaver as a realistic possibility. Piercing Light gives him a 5 second, 1.2x AD ratio spammable ability that can do serious group damage late game. Combined with a Black Cleaver, this can be spammed frequently and liberally to shred the opponent's armor and set up a culling that will quickly be dealing nearly unmitigated damage.
Clearly, with all the AP ratios, Riot wants him to be looking at Tri-Force as well. The changes to Tri-Force make it a much more attractive item to buy and the movement speed inherent in the item along with the altered passive buff could make Lucian the skirmisher that Quinn was supposed to be.
Because a fair amount of his damage comes from his passive, Piercing Light and his ult, he's somewhat less attack speed reliant than other champs. While BotRK isn't a "bad" item on him, it also isn't as amazing on him as it is on someone like Vayne or even Tristana.
Good supports and matchups
With good movement and escapes, Lucian doesn't have some of the typical holes in his arsenal that champs like Miss Fortune sometimes suffer from. He has strong trading potential with the right timing, and champs with good CC (Taric, Nami, Leona) make strong partnership candidates. Alternatively, he appears to be one of the better lane pushers in the game (especially if his passive double attacks turrets), which would lead to potential with someone like Lulu as a strong early game lane pushing tandem.
Expect Caitlyn to be a strong counter to him due to her range advantage, especially in the next week while people get a feel for him. A good Vayne will be difficult to trade with because of her ability to tumble away from the Q, though you might be able to use your Q to force a tumble, wait for her auto-attack and then trade aggressively in that small window. If she can't get her third AA on you, you likely come out ahead.
On the other hand, Graves and MF are probably pretty good matchups, as Lucian's ability to dash in for a trade appears to be stronger than theirs (Dash, double attack, W, Double attack gaining speed, walk away and Q, double attack to finish the trade).
All of these matchups are theoretical, at best. Lucian is rhythm-based champion, and getting the most out of his passive and Culling is going to take some serious time. When laning against him, remember that a stun will interrupt his Ultimate and that his Q has a relatively short range that is easy to bait and dodge by moving perpendicular, since he cannot lead you with it. His W is unlikely to get leveled past level one, so your trading window is after you see the Q fire or if you force a Relentless Pursuit.