The preamble to the LGPLv2.1 has a good warning:
Pay close attention to the difference between a "work based on the
library" and a "work that uses the library". The former contains code
derived from the library, whereas the latter must be combined with the
library in order to run.
If an Apache 2.0-licensed software is linked to and is dependent on a LGPLv2.1-licensed library, then it's a "work that uses the library", which is not subject to the terms and conditions of the LGPLv2.1, according to section 6 of the LGPLv2.1:
As an exception to the Sections above, you may also combine or link a
"work that uses the Library" with the Library to produce a work
containing portions of the Library, and distribute that work under
terms of your choice, provided that the terms permit modification of
the work for the customer's own use and reverse engineering for
debugging such modifications.
If a LGPLv2.1-licensed software or library is linked to and is dependent on a Apache 2.0-licensed library, then the combination of the two is a "work based on the library", and is therefore subject to section 2 of the LGPLv2.1 which states that the combination or the whole, is licensed under LGPLv2.1:
But when you distribute the same sections as part of a whole which is
a work based on the Library, the distribution of the whole must be on
the terms of this License, whose permissions for other licensees
extend to the entire whole, and thus to each and every part regardless
of who wrote it.
The Apache 2.0-licensed library itself, when used in this latter combination, would be licensed under the terms of both the Apache 2.0 License AND the LGPLv2.1 simultaneously.
The Apache 2.0 License imposes patent-related terms and conditions (in sections 3 and 9) which the LGPLv2.1 does not have.
Section 10 of the LGPLv2.1 states:
You may not impose any further restrictions on the recipients'
exercise of the rights granted herein.
So, the Apache 2.0 License conflicts with LGPLv2.1, because Apache 2.0 has additional terms and conditions on patent, while LGPLv2.1 does not allow imposition of further restrictions.
This is why a LGPLv2.1-licensed software or library cannot be linked to and be dependent on a Apache 2.0-licensed library, unless the LGPLv2.1-licensed software or library is licensed as "LGPL version 2.1 or later", in which case the combination can be relicensed under LGPLv3 or GPLv3. The Apache 2.0 License is compatible with LGPLv3 or GPLv3.
If the LGPLv2.1 software or library is licensed as "LGPL version 2.1 only" (and no later), then the combination can be relicensed under GPLv3 based on Section 3 of the LGPLv2.1. The Apache 2.0 License is compatible with GPLv3.