Tilting modules The theory of tilting modules seems to be a very fruitful field. I have some questions which seem natural to me, but can be trivial or stupid for people who works with this sort of theory.
1) Do all categories of modules with a partial order in the set of indexes for the simple modules have tilting modules?
2) If a category has the simple modules being standard modules and the tilting modules as injective hulls of simple modules, then can we say something interesting about this category? Is it a distinguished category in any sense?
3) Why is important to construct tilting modules?
 A: 1) As already remarked in the comments this depends very much on the definition of tilting module you have.
a) You said that for you a tilting module is one having a standard and a costandard filtration. So let's say for you a standard module is a factor module of an indecomposable projective module modulo all composition factors being greater than the top of that module. In that case not all algebras have tilting modules in the above sense. For example consider the local algebra $\mathbb{C}[X,Y]/(X^2,Y^2,XY)$. Then the standard module is the indecomposable projective, the costandardmodule is the indecomposable injective. Both are different and their additive hull forms the category of $\Delta$-filtered, respectively $\nabla$-filtered modules. Tilting modules in that sense occur for quasi-hereditary algebras and certain generalisations, e.g. standardly stratified algebras. In the setting of quasi-hereditary algebras, there are the socalled tilting modules, one for each vertex $T(i)$ and I think the name comes from the fact that $\bigoplus_{i=1}^n T(i)$ forms a tilting module (called the characteristic tilting module) in the more generally used context Mariano wrote about in the comments. If you want to read more about quasi-hereditary algebras, I suggest reading the two articles: 


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*[Dlab, Ringel: The module theoretic approach to quasi-hereditary algebras] or 

*[Klucznik, Koenig: Characteristic tilting modules over quasi-hereditary algebras]


b) A (generalised) tilting module over a finite dimensional algebra $A$ is a module $T$ such that:


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*$T$ has finite projective dimension

*$\operatorname{Ext}^i(T,T)=0$ for all $i>0$

*There is an exact sequence $0\to A\to T_1\to \dots\to T_n\to 0$ where $T_i\in \operatorname{add} T$.


Those modules exist for every finite dimensional algebra $A$ (but do not need to have anything to do with the partial order). For example (as Mariano wrote) the direct sum of all indecomposable projectives is such a module. In fact one can show (as for this example and the characteristic tilting module) that there are exactly as many indecomposable direct summands up to isomorphism as there are simples. Those tilting modules (and this partly answers 3)) are important because they yield derived equivalences between module categories (that of $A$ and that of $\operatorname{End}_A(T)^{op}$. (If you don't know this equivalence is much weaker than "Morita equivalence" which tells you that the module categories are equivalent). A good point to start with when reading about this type of tilting modules is maybe [Assem, Simson, Skowronski: Elements of the Representation theory of finite dimensional algebras: Volume 1, Chapter VI: Tilting theory]
In the special context of quasi-hereditary algebras you also get an equivalence between certain subcategories, those of $\Delta$-filtered modules for $A$ and those of $\nabla$-filtered modules for $R(A)=\operatorname{End}_A(T)^{op}$, the Ringel dual of $A$.
2) A finite dimensional algebra having simple standard modules (and then also injective tilting modules) is sometimes called (note that the claim in the comment is false) a directed algebra (because its quiver is a directed graph, i.e. only having arrows in increasing direction). Those algebras are quite important for the theory of quasi-hereditary algebras in general. They are somehow the associative-algebra analogue of solvable Lie algebras. There is the notion of an exact Borel subalgebra by Koenig. A good starting point to read (although with a false claim in one of the examples) is:


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*[Koenig: Exact Borel subalgebras of quasi-hereditary algebras]


3) Already answered in 1b)
