I just started watching Fairy Tail and I got to know about the dragon slayers Natsu and Gajeel.
However, what actually are dragon slayers? If they did kill a dragon, why does Natsu seem to be so attached to Igneel?
Short answer: First generation dragon slayers are enchanted to slowly physically and magically develop dragon-like traits(ex. biological immortality). So they are part dragon/ part human with the ability to use both human and dragon magics. Second generations can use dragon magic but always were and will be completely human(ex. not biologically immortal).
Much more long winded and spoiling answer: In this series dragons function a lot like wizards operating on a massive and sometimes literally inconceivably massive power scale. Humans being so fundamentally different from dragons means that humans, by default, are literally incapable of using the same type of magic as a dragon. The glory of being a dragon slayer is that you have the power to use dragon magic despite being human. Second generation dragon slayers are given the power to use dragon magic through the implantation of a dragon lacrima while first generations were physically changed by the dragons themselves. The main advantage of being a first generation vs a second generation is that first generations slowly develop the same power level as a dragon, not just the same type of magic. The dragon force is the beginning of this process and is the main reason that dragon slayer magic is famous for being powerful. Full grown dragon slayers are akin to walking reusable etherion blasts. Throughout the series they never die except from other full grown dragon slayers(including suicides); although, logic says a powerful enough dragon should have the power to kill one as well. They also never age so older ones tend to have enormous experience/talent using their magic.
At your current place in the story, you should understand the following: Dragon Slayer magic is a rare type of magic taught to humans by dragons. This gives them some kind of elemental magic such as fire/metal/sound/light/water/sky/etc. This magic should be particularly good at fighting other dragons. The dragons disappeared many years ago. Specifically, Natsu was taught Dragon Slayer magic by Igneel who he is attached to like a child/parent. Igneel also seemed to disappear without warning or explanation. Wouldn't you look for your missing parent/friend/mentor?
Where you are now, there are apparent contradictions in this narrative. The exact nature of what is going on is important to the story going forward and it would be best to learn it as you watch the anime or read the manga (assuming you don't just want to skip to the end).
There are fake dragon slayers whose powers are derived from implanted Dragon Lacrymas. These slayers likely never met a dragon.
It is not necessary to kill a dragon in order to become a dragon slayers. They are called dragon slayers because they have very rare and very powerful magic which can only be taught by dragon itself.
Igneel is only dragon natsu knew since he was a kid so he is attached to him and he also admires him. You can say that he is only parent for natsu. All dragons are disappeared many years ago and there are many mysteries behind their disappearance.
You said that you just started watching fairy tail so there are more suspense about the origin of dragon slayers is waiting for you as you continue the series.