For references, probably you already know things like Kollar-Mori and Kollar's "Singularities of pairs", for LC-centers and subadjunction, Kollar has some notes on that as well. There's some sections on this in the recent book by Hacon-Kovacs too that I've looked at recently.

With regards to your specific questions I have some comments but probably VA will have better comments:

1. an LC pair is much more difficult to treat than a DLT pair

The archetypical LC pair is probably $(\mathbb{A^2}, Div(x) + Div(y))$, the two coordinate axes. This is a simple normal crossings pair and we understand how they behave pretty well with respect to numerous operations. On the other hand, if you have a pair $(X, D)$ where the pair is klt, then even if the singularities of $D$ are bad, the fact that $(X, D)$ is klt means that you can perturb the coefficients of $D$ in many ways. This gives one a great amount of flexibility in numerous situations. DLT, by Szabo's criterion, is a combination of these notions. Basically, it is LC and where it is not KLT, it is SNC (simply normal crossings), which we understand. Dually, where it's not SNC, it's KLT, and we can perturb things and do those other tricks.

2. what the LC centers of a pair are and why they are "special" subvarieties for a pair.

The most basic example of a LC-center of a pair $(X, D)$ is a prime divisor $D_i$ which is a component of $D$ and such that the $D_i$-coefficient of $D$ is $1$. For example, if $X$ is smooth, and $D$ is a prime divisor, then $D$ is a LC-center of $(X, D)$. Why is this nice? This lets us relate the canonical divisor $K_X + D$ of $(X, D)$ and the canonical divisor of $D$.
Explicitly, in this smooth case, $(K_X + D)|_D = K_D$. Because of this, you can translate many properties of the pair $(X, D)$ to things on $D$ (look up extension theorems for example, or adjunction/inversion of adjunction, variants of this hold without the smooth hypothesis). This is very very useful for induction on dimension. LC-centers are a way to generalize this to higher codimensional subvarieties.

By definition, LC-centers are exactly the images of divisors $D_i'$ where $D_i'$ is a component of $D'$ with coefficient $1$, and $(X', D')$ is a pair obtained by taking a log resolution $\pi$ of $(X, D)$ and setting $D' = \pi^*(K_X + D) - K_{X'}$.

In particular, if $W$ is an LC-center of a pair $(X, D)$, then one has $$(K_X + D)|_W = K_W + (\text{some correction terms}).$$ Again, many properties of the pair $(X, D)$ can be translated to properties of $(W, (\text{correction terms}) )$, this isn't trivial to show.

One should note that even if you have a 1-dimensional LC-center, then there can still be a correction factor (just not in the smooth example I described above).

For references, probably you already know things like Kollar-Mori and Kollar's "Singularities of pairs", for LC-centers and subadjunction, Kollar has some notes on that as well. There's some sections on this in the recent book by Hacon-Kovacs too that I've looked at recently.

With regards to your specific questions I have some comments but probably VA will have better comments:

1. an LC pair is much more difficult to treat than a DLT pair

The archetypical LC pair is probably $(\mathbb{A^2}, Div(x) + Div(y))$, the two coordinate axes (EDIT: this example is also DLT, and doesn't really represent the most poorly-behaved properties of LC-pairs). This is a simple normal crossings (SNC) pair and we understand how they behave pretty well with respect to numerous operations. On the other hand, if you have a pair $(X, D)$ where the pair is KLT, then even if the singularities of $D$ are bad, the fact that $(X, D)$ is klt means that you can perturb the coefficients of $D$ in many ways. This gives one a great amount of flexibility in numerous situations.

DLT, by Szabo's criterion, is a combination of KLT and SNC with coefficients $\leq 1$. Basically, a DLT pair is LC and where it is not KLT, it is SNC (simply normal crossings), which we understand. Dually, where it's not SNC, it's KLT, and we can perturb things and do those other tricks.

2. what the LC centers of a pair are and why they are "special" subvarieties for a pair.

The most basic example of a LC-center of a pair $(X, D)$ is a prime divisor $D_i$ which is a component of $D$ and such that the $D_i$-coefficient of $D$ is $1$. For example, if $X$ is smooth, and $D$ is a prime divisor, then $D$ is a LC-center of $(X, D)$. Why is this nice? This lets us relate the canonical divisor $K_X + D$ of $(X, D)$ and the canonical divisor of $D$.
Explicitly, in this smooth case, $(K_X + D)|_D = K_D$. Because of this, you can translate many properties of the pair $(X, D)$ to things on $D$ (look up extension theorems for example, or adjunction/inversion of adjunction, variants of this hold without the smooth hypothesis). This is very very useful for induction on dimension. LC-centers are a way to generalize this to higher codimensional subvarieties.

By definition, LC-centers are exactly the images of divisors $D_i'$ where $D_i'$ is a component of $D'$ with coefficient $1$, and $(X', D')$ is a pair obtained by taking a log resolution $\pi$ of $(X, D)$ and setting $D' = \pi^*(K_X + D) - K_{X'}$.

In particular, if $W$ is an LC-center of a pair $(X, D)$, then one has $$(K_X + D)|_W = K_W + (\text{some correction terms}).$$ Again, many properties of the pair $(X, D)$ can be translated to properties of $(W, (\text{correction terms}) )$, this isn't trivial to show.

One should note that even if you have a 1-dimensional LC-center, then there can still be a correction factor (just not in the smooth example I described above).

For references, probably you already know things like Kollar-Mori and Kollar's "Singularities of pairs", for LC-centers and subadjunction, Kollar has some notes on that as well. There's some sections on this in the recent book by Hacon-Kovacs too that I've looked at.

With regards to your specific questions I have some comments but probably VA will have better comments:

1. an LC pair is much more difficult to treat than a DLT pair

The archetypical LC pair is probably $(\mathbb{A^2}, Div(x) + Div(y))$, the two coordinate axes. This is a simple normal crossings pair and we understand how they behave pretty well with respect to numerous operations. On the other hand, if you have a pair $(X, D)$ where the pair is klt, then even if the singularities of $D$ are bad, the fact that $(X, D)$ is klt means that you can perturb the coefficients of $D$ in many ways. This gives one a great amount of flexibility in numerous situations. DLT, by Szabo's criterion, is a combination of these notions. Basically, it is LC and where it is not KLT, it is SNC (simply normal crossings), which we understand. Dually, where it's not SNC, it's KLT, and we can perturb things and do those other tricks.

2. what the LC centers of a pair are and why they are "special" subvarieties for a pair.

The most basic example of a LC-center of a pair $(X, D)$ is a prime divisor $D_i$ which is a component of $D$ and such that the $D_i$-coefficient of $D$ is $1$. For example, if $X$ is smooth, and $D$ is a prime divisor, then $D$ is a LC-center of $(X, D)$. Why is this nice? This lets us relate the canonical divisor $K_X + D$ of $(X, D)$ and the canonical divisor of $D$.
Explicitly $(K_X + D)|_D = K_D$. Because of this, you can translate many properties of the pair $(X, D)$ to things on $D$ (look up extension theorems for example, or adjunction/inversion of adjunction). This is very very useful for induction on dimension. LC-centers are a way to generalize this to higher codimensional subvarieties.

In particular, if $W$ is an LC-center of a pair $(X, D)$, then one has $$(K_X + D)|_W = K_W + (\text{some correction terms})$$. Again, many properties of the pair $(X, D)$ can be translated to properties of $(W, (\text{correction terms}) )$. One should note that even if you have a 1-dimensional LC-center, then there can still be a correction factor (just not in the smooth example I described above).

For references, probably you already know things like Kollar-Mori and Kollar's "Singularities of pairs", for LC-centers and subadjunction, Kollar has some notes on that as well. There's some sections on this in the recent book by Hacon-Kovacs too that I've looked at recently.

With regards to your specific questions I have some comments but probably VA will have better comments:

1. an LC pair is much more difficult to treat than a DLT pair

The archetypical LC pair is probably $(\mathbb{A^2}, Div(x) + Div(y))$, the two coordinate axes. This is a simple normal crossings pair and we understand how they behave pretty well with respect to numerous operations. On the other hand, if you have a pair $(X, D)$ where the pair is klt, then even if the singularities of $D$ are bad, the fact that $(X, D)$ is klt means that you can perturb the coefficients of $D$ in many ways. This gives one a great amount of flexibility in numerous situations. DLT, by Szabo's criterion, is a combination of these notions. Basically, it is LC and where it is not KLT, it is SNC (simply normal crossings), which we understand. Dually, where it's not SNC, it's KLT, and we can perturb things and do those other tricks.

2. what the LC centers of a pair are and why they are "special" subvarieties for a pair.

The most basic example of a LC-center of a pair $(X, D)$ is a prime divisor $D_i$ which is a component of $D$ and such that the $D_i$-coefficient of $D$ is $1$. For example, if $X$ is smooth, and $D$ is a prime divisor, then $D$ is a LC-center of $(X, D)$. Why is this nice? This lets us relate the canonical divisor $K_X + D$ of $(X, D)$ and the canonical divisor of $D$.
Explicitly, in this smooth case, $(K_X + D)|_D = K_D$. Because of this, you can translate many properties of the pair $(X, D)$ to things on $D$ (look up extension theorems for example, or adjunction/inversion of adjunction, variants of this hold without the smooth hypothesis). This is very very useful for induction on dimension. LC-centers are a way to generalize this to higher codimensional subvarieties.

By definition, LC-centers are exactly the images of divisors $D_i'$ where $D_i'$ is a component of $D'$ with coefficient $1$, and $(X', D')$ is a pair obtained by taking a log resolution $\pi$ of $(X, D)$ and setting $D' = \pi^*(K_X + D) - K_{X'}$.

In particular, if $W$ is an LC-center of a pair $(X, D)$, then one has $$(K_X + D)|_W = K_W + (\text{some correction terms}).$$ Again, many properties of the pair $(X, D)$ can be translated to properties of $(W, (\text{correction terms}) )$, this isn't trivial to show.

One should note that even if you have a 1-dimensional LC-center, then there can still be a correction factor (just not in the smooth example I described above).