Generally using regexen for parsing markup is tricky and best left to modules such as HTML::TreeParser. If you can absolutely predict what the markup will be then you may get away with the following:

use warnings;
use strict;

my $str = <<TXT;
Nerve cells come in many shapes and sizes, but they all have a number 
+of
identifiable parts. A typical nerve cell is shown in <figr n="1">Figur
+e
1</figr>. Like all other cells in the body, it has a nucleus that cont
+ains
genetic information.<figr n="2">Figure 2</figr>. The cell is covered b
+y a
membrane and is filled with a fluid.
TXT

$str =~ s/
            <figr\sn="(\d+?)"> # tag including figure number
            (.*?)              # element contents
            <\/figr>           # close tag
        /
            "<FIGIND NUM=\"$1\" ID=\"FG." . # replacement tag
            sprintf("%03d", $1) .           # padded number
            "\">$2<\/FIGIND>"               # remainder of element
        /gmsxe; # Global, multi-line, ignore newline, ignore whitespac
+e, evaluate
print $str;
[download]

Prints:

Nerve cells come in many shapes and sizes, but they all have a number 
+of
identifiable parts. A typical nerve cell is shown in <FIGIND NUM="1" I
+D="FG.001">Figure
1</FIGIND>. Like all other cells in the body, it has a nucleus that co
+ntains
genetic information.<FIGIND NUM="2" ID="FG.002">Figure 2</FIGIND>. The
+ cell is covered by a
membrane and is filled with a fluid.
[download]