  We show that many large cardinal notions can be characterized in terms of the
existence of certain elementary embeddings between transitive set-sized
structures, that map their critical point to the large cardinal in question. In
particular, we provide such embedding characterizations also for several large
cardinal notions for which no embedding characterizations have been known so
far, namely for subtle, for ineffable, and for $\lambda$-ineffable cardinals.
As an application, which we will study in detail in a subsequent paper, we
present the basic idea of our concept of internal large cardinals. We provide
the definition of certain kinds of internally subtle, internally
$\lambda$-ineffable and internally supercompact cardinals, and show that these
correspond to generalized tree properties, that were investigated by Wei\ss\ in
his [16] and [17], and by Viale and Wei\ss\ in [15]. In particular, this yields
new proofs of Wei\ss 's results from [16] and [17], eliminating problems
contained in the original proofs.
